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Solving Quadratic Equations by Taking Square Roots Worksheets

Unlike the standard form: ax 2 + bx + c = 0, most of the quadratic equations offered in this pack of printable high school worksheets have no middle term. Such equations are known as pure quadratic equations and are of the form ax 2 - c = 0. Solving these quadratic equations is made a lot easier by by taking square roots. Let's now get into the process! Here’s all you have to do. Isolate the leading term on the left-hand side of the equation and the constant term on the right-hand side, take square roots on both sides, and simplify both sides for the values of x. The equations that have the middle term can also be solved by finding square roots. Yet again, the middle term is excluded by using the appropriate algebraic identities. Get started with our free worksheet!

Solve Quadratic Equations by Taking Square Roots - Level 1

Solve Quadratic Equations by Taking Square Roots - Level 1

Push-start your practice of finding the real and complex roots of quadratic equations with this set of pdf worksheets presenting 30 pure quadratic equations. Note that the coefficient of the leading term is 1 in every equation. Hence, simply rewrite the given equation in the form of x 2 = c, and proceed to solve for x.

  • Download the set

Solve Quadratic Equations by Taking Square Roots - Level 2

Solve Quadratic Equations by Taking Square Roots - Level 2

Keep one jump ahead of your peers with this printable practice set perfect for high school students! The quadratic equations here involve integers and fractions. You need to rewrite the equation to the desired form, isolate the x 2 term, take square roots, and perform simplification on both sides.

Related Worksheets

» Solving Equations | Completing the Squares

» Solving Quadratic Equations | Factoring

» Solving Equations | Quadratic Formula

» Sum and Product of the Roots

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  • 9.1 Solve Quadratic Equations Using the Square Root Property
  • Introduction
  • 1.1 Use the Language of Algebra
  • 1.2 Integers
  • 1.3 Fractions
  • 1.4 Decimals
  • 1.5 Properties of Real Numbers
  • Key Concepts
  • Review Exercises
  • Practice Test
  • 2.1 Use a General Strategy to Solve Linear Equations
  • 2.2 Use a Problem Solving Strategy
  • 2.3 Solve a Formula for a Specific Variable
  • 2.4 Solve Mixture and Uniform Motion Applications
  • 2.5 Solve Linear Inequalities
  • 2.6 Solve Compound Inequalities
  • 2.7 Solve Absolute Value Inequalities
  • 3.1 Graph Linear Equations in Two Variables
  • 3.2 Slope of a Line
  • 3.3 Find the Equation of a Line
  • 3.4 Graph Linear Inequalities in Two Variables
  • 3.5 Relations and Functions
  • 3.6 Graphs of Functions
  • 4.1 Solve Systems of Linear Equations with Two Variables
  • 4.2 Solve Applications with Systems of Equations
  • 4.3 Solve Mixture Applications with Systems of Equations
  • 4.4 Solve Systems of Equations with Three Variables
  • 4.5 Solve Systems of Equations Using Matrices
  • 4.6 Solve Systems of Equations Using Determinants
  • 4.7 Graphing Systems of Linear Inequalities
  • 5.1 Add and Subtract Polynomials
  • 5.2 Properties of Exponents and Scientific Notation
  • 5.3 Multiply Polynomials
  • 5.4 Dividing Polynomials
  • Introduction to Factoring
  • 6.1 Greatest Common Factor and Factor by Grouping
  • 6.2 Factor Trinomials
  • 6.3 Factor Special Products
  • 6.4 General Strategy for Factoring Polynomials
  • 6.5 Polynomial Equations
  • 7.1 Multiply and Divide Rational Expressions
  • 7.2 Add and Subtract Rational Expressions
  • 7.3 Simplify Complex Rational Expressions
  • 7.4 Solve Rational Equations
  • 7.5 Solve Applications with Rational Equations
  • 7.6 Solve Rational Inequalities
  • 8.1 Simplify Expressions with Roots
  • 8.2 Simplify Radical Expressions
  • 8.3 Simplify Rational Exponents
  • 8.4 Add, Subtract, and Multiply Radical Expressions
  • 8.5 Divide Radical Expressions
  • 8.6 Solve Radical Equations
  • 8.7 Use Radicals in Functions
  • 8.8 Use the Complex Number System
  • 9.2 Solve Quadratic Equations by Completing the Square
  • 9.3 Solve Quadratic Equations Using the Quadratic Formula
  • 9.4 Solve Equations in Quadratic Form
  • 9.5 Solve Applications of Quadratic Equations
  • 9.6 Graph Quadratic Functions Using Properties
  • 9.7 Graph Quadratic Functions Using Transformations
  • 9.8 Solve Quadratic Inequalities
  • 10.1 Finding Composite and Inverse Functions
  • 10.2 Evaluate and Graph Exponential Functions
  • 10.3 Evaluate and Graph Logarithmic Functions
  • 10.4 Use the Properties of Logarithms
  • 10.5 Solve Exponential and Logarithmic Equations
  • 11.1 Distance and Midpoint Formulas; Circles
  • 11.2 Parabolas
  • 11.3 Ellipses
  • 11.4 Hyperbolas
  • 11.5 Solve Systems of Nonlinear Equations
  • 12.1 Sequences
  • 12.2 Arithmetic Sequences
  • 12.3 Geometric Sequences and Series
  • 12.4 Binomial Theorem

Learning Objectives

By the end of this section, you will be able to:

  • Solve quadratic equations of the form a x 2 = k a x 2 = k using the Square Root Property
  • Solve quadratic equations of the form a ( x – h ) 2 = k a ( x – h ) 2 = k using the Square Root Property

Be Prepared 9.1

Before you get started, take this readiness quiz.

Simplify: 128 . 128 . If you missed this problem, review Example 8.13 .

Be Prepared 9.2

Simplify: 32 5 32 5 . If you missed this problem, review Example 8.50 .

Be Prepared 9.3

Factor: 9 x 2 − 12 x + 4 9 x 2 − 12 x + 4 . If you missed this problem, review Example 6.23 .

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a ≠ 0 a ≠ 0 . Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax 2 . We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.

Solve Quadratic Equations of the form a x 2 = k a x 2 = k using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x 2 = 9.

We can easily use factoring to find the solutions of similar equations, like x 2 = 16 and x 2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, x = 4 , x = −4 x = 4 , x = −4 and x = 5 , x = −5 . x = 5 , x = −5 .

But what happens when we have an equation like x 2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13) 2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:

Since these equations are all of the form x 2 = k , the square root definition tells us the solutions are the two square roots of k . This leads to the Square Root Property .

Square Root Property

If x 2 = k , then

Notice that the Square Root Property gives two solutions to an equation of the form x 2 = k , the principal square root of k k and its opposite. We could also write the solution as x = ± k . x = ± k . We read this as x equals positive or negative the square root of k .

Now we will solve the equation x 2 = 9 again, this time using the Square Root Property.

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x 2 = 7.

We cannot simplify 7 7 , so we leave the answer as a radical.

Example 9.1

How to solve a quadratic equation of the form ax 2 = k using the square root property.

Solve: x 2 − 50 = 0 . x 2 − 50 = 0 .

Solve: x 2 − 48 = 0 . x 2 − 48 = 0 .

Solve: y 2 − 27 = 0 . y 2 − 27 = 0 .

The steps to take to use the Square Root Property to solve a quadratic equation are listed here.

Solve a quadratic equation using the square root property.

  • Step 1. Isolate the quadratic term and make its coefficient one.
  • Step 2. Use Square Root Property.
  • Step 3. Simplify the radical.
  • Step 4. Check the solutions.

In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.

Example 9.2

Solve: 3 z 2 = 108 . 3 z 2 = 108 .

Solve: 2 x 2 = 98 . 2 x 2 = 98 .

Solve: 5 m 2 = 80 . 5 m 2 = 80 .

The Square Root Property states ‘If x 2 = k x 2 = k ,’ What will happen if k < 0 ? k < 0 ? This will be the case in the next example.

Example 9.3

Solve: x 2 + 72 = 0 x 2 + 72 = 0 .

Solve: c 2 + 12 = 0 . c 2 + 12 = 0 .

Solve: q 2 + 24 = 0 . q 2 + 24 = 0 .

Our method also works when fractions occur in the equation; we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.

Example 9.4

Solve: 2 3 u 2 + 5 = 17 . 2 3 u 2 + 5 = 17 .

Solve: 1 2 x 2 + 4 = 24 . 1 2 x 2 + 4 = 24 .

Solve: 3 4 y 2 − 3 = 18 . 3 4 y 2 − 3 = 18 .

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator .

Example 9.5

Solve: 2 x 2 − 8 = 41 . 2 x 2 − 8 = 41 .

Solve: 5 r 2 − 2 = 34 . 5 r 2 − 2 = 34 .

Try It 9.10

Solve: 3 t 2 + 6 = 70 . 3 t 2 + 6 = 70 .

Solve Quadratic Equations of the Form a ( x − h ) 2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation of the form a ( x − h ) 2 = k as well. Notice that the quadratic term, x , in the original form ax 2 = k is replaced with ( x − h ).

The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a , then the Square Root Property can be used on ( x − h ) 2 .

Example 9.6

Solve: 4 ( y − 7 ) 2 = 48 . 4 ( y − 7 ) 2 = 48 .

Try It 9.11

Solve: 3 ( a − 3 ) 2 = 54 . 3 ( a − 3 ) 2 = 54 .

Try It 9.12

Solve: 2 ( b + 2 ) 2 = 80 . 2 ( b + 2 ) 2 = 80 .

Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

Example 9.7

Solve: ( x − 1 3 ) 2 = 5 9 . ( x − 1 3 ) 2 = 5 9 .

Try It 9.13

Solve: ( x − 1 2 ) 2 = 5 4 . ( x − 1 2 ) 2 = 5 4 .

Try It 9.14

Solve: ( y + 3 4 ) 2 = 7 16 . ( y + 3 4 ) 2 = 7 16 .

We will start the solution to the next example by isolating the binomial term.

Example 9.8

Solve: 2 ( x − 2 ) 2 + 3 = 57 . 2 ( x − 2 ) 2 + 3 = 57 .

Try It 9.15

Solve: 5 ( a − 5 ) 2 + 4 = 104 . 5 ( a − 5 ) 2 + 4 = 104 .

Try It 9.16

Solve: 3 ( b + 3 ) 2 − 8 = 88 . 3 ( b + 3 ) 2 − 8 = 88 .

Sometimes the solutions are complex numbers.

Example 9.9

Solve: ( 2 x − 3 ) 2 = −12 . ( 2 x − 3 ) 2 = −12 .

Try It 9.17

Solve: ( 3 r + 4 ) 2 = −8 . ( 3 r + 4 ) 2 = −8 .

Try It 9.18

Solve: ( 2 t − 8 ) 2 = −10 . ( 2 t − 8 ) 2 = −10 .

The left sides of the equations in the next two examples do not seem to be of the form a ( x − h ) 2 . But they are perfect square trinomials, so we will factor to put them in the form we need.

Example 9.10

Solve: 4 n 2 + 4 n + 1 = 16 . 4 n 2 + 4 n + 1 = 16 .

We notice the left side of the equation is a perfect square trinomial. We will factor it first.

Try It 9.19

Solve: 9 m 2 − 12 m + 4 = 25 . 9 m 2 − 12 m + 4 = 25 .

Try It 9.20

Solve: 16 n 2 + 40 n + 25 = 4 . 16 n 2 + 40 n + 25 = 4 .

Access this online resource for additional instruction and practice with using the Square Root Property to solve quadratic equations.

  • Solving Quadratic Equations: The Square Root Property
  • Using the Square Root Property to Solve Quadratic Equations

Section 9.1 Exercises

Practice makes perfect.

Solve Quadratic Equations of the Form ax 2 = k Using the Square Root Property

In the following exercises, solve each equation.

a 2 = 49 a 2 = 49

b 2 = 144 b 2 = 144

r 2 − 24 = 0 r 2 − 24 = 0

t 2 − 75 = 0 t 2 − 75 = 0

u 2 − 300 = 0 u 2 − 300 = 0

v 2 − 80 = 0 v 2 − 80 = 0

4 m 2 = 36 4 m 2 = 36

3 n 2 = 48 3 n 2 = 48

4 3 x 2 = 48 4 3 x 2 = 48

5 3 y 2 = 60 5 3 y 2 = 60

x 2 + 25 = 0 x 2 + 25 = 0

y 2 + 64 = 0 y 2 + 64 = 0

x 2 + 63 = 0 x 2 + 63 = 0

y 2 + 45 = 0 y 2 + 45 = 0

4 3 x 2 + 2 = 110 4 3 x 2 + 2 = 110

2 3 y 2 − 8 = −2 2 3 y 2 − 8 = −2

2 5 a 2 + 3 = 11 2 5 a 2 + 3 = 11

3 2 b 2 − 7 = 41 3 2 b 2 − 7 = 41

7 p 2 + 10 = 26 7 p 2 + 10 = 26

2 q 2 + 5 = 30 2 q 2 + 5 = 30

5 y 2 − 7 = 25 5 y 2 − 7 = 25

3 x 2 − 8 = 46 3 x 2 − 8 = 46

( u − 6 ) 2 = 64 ( u − 6 ) 2 = 64

( v + 10 ) 2 = 121 ( v + 10 ) 2 = 121

( m − 6 ) 2 = 20 ( m − 6 ) 2 = 20

( n + 5 ) 2 = 32 ( n + 5 ) 2 = 32

( r − 1 2 ) 2 = 3 4 ( r − 1 2 ) 2 = 3 4

( x + 1 5 ) 2 = 7 25 ( x + 1 5 ) 2 = 7 25

( y + 2 3 ) 2 = 8 81 ( y + 2 3 ) 2 = 8 81

( t − 5 6 ) 2 = 11 25 ( t − 5 6 ) 2 = 11 25

( a − 7 ) 2 + 5 = 55 ( a − 7 ) 2 + 5 = 55

( b − 1 ) 2 − 9 = 39 ( b − 1 ) 2 − 9 = 39

4 ( x + 3 ) 2 − 5 = 27 4 ( x + 3 ) 2 − 5 = 27

5 ( x + 3 ) 2 − 7 = 68 5 ( x + 3 ) 2 − 7 = 68

( 5 c + 1 ) 2 = −27 ( 5 c + 1 ) 2 = −27

( 8 d − 6 ) 2 = −24 ( 8 d − 6 ) 2 = −24

( 4 x − 3 ) 2 + 11 = −17 ( 4 x − 3 ) 2 + 11 = −17

( 2 y + 1 ) 2 − 5 = −23 ( 2 y + 1 ) 2 − 5 = −23

m 2 − 4 m + 4 = 8 m 2 − 4 m + 4 = 8

n 2 + 8 n + 16 = 27 n 2 + 8 n + 16 = 27

x 2 − 6 x + 9 = 12 x 2 − 6 x + 9 = 12

y 2 + 12 y + 36 = 32 y 2 + 12 y + 36 = 32

25 x 2 − 30 x + 9 = 36 25 x 2 − 30 x + 9 = 36

9 y 2 + 12 y + 4 = 9 9 y 2 + 12 y + 4 = 9

36 x 2 − 24 x + 4 = 81 36 x 2 − 24 x + 4 = 81

64 x 2 + 144 x + 81 = 25 64 x 2 + 144 x + 81 = 25

Mixed Practice

In the following exercises, solve using the Square Root Property.

2 r 2 = 32 2 r 2 = 32

4 t 2 = 16 4 t 2 = 16

( a − 4 ) 2 = 28 ( a − 4 ) 2 = 28

( b + 7 ) 2 = 8 ( b + 7 ) 2 = 8

9 w 2 − 24 w + 16 = 1 9 w 2 − 24 w + 16 = 1

4 z 2 + 4 z + 1 = 49 4 z 2 + 4 z + 1 = 49

a 2 − 18 = 0 a 2 − 18 = 0

b 2 − 108 = 0 b 2 − 108 = 0

( p − 1 3 ) 2 = 7 9 ( p − 1 3 ) 2 = 7 9

( q − 3 5 ) 2 = 3 4 ( q − 3 5 ) 2 = 3 4

m 2 + 12 = 0 m 2 + 12 = 0

n 2 + 48 = 0 . n 2 + 48 = 0 .

u 2 − 14 u + 49 = 72 u 2 − 14 u + 49 = 72

v 2 + 18 v + 81 = 50 v 2 + 18 v + 81 = 50

( m − 4 ) 2 + 3 = 15 ( m − 4 ) 2 + 3 = 15

( n − 7 ) 2 − 8 = 64 ( n − 7 ) 2 − 8 = 64

( x + 5 ) 2 = 4 ( x + 5 ) 2 = 4

( y − 4 ) 2 = 64 ( y − 4 ) 2 = 64

6 c 2 + 4 = 29 6 c 2 + 4 = 29

2 d 2 − 4 = 77 2 d 2 − 4 = 77

( x − 6 ) 2 + 7 = 3 ( x − 6 ) 2 + 7 = 3

( y − 4 ) 2 + 10 = 9 ( y − 4 ) 2 + 10 = 9

Writing Exercises

In your own words, explain the Square Root Property.

In your own words, explain how to use the Square Root Property to solve the quadratic equation ( x + 2 ) 2 = 16 ( x + 2 ) 2 = 16 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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  • Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/9-1-solve-quadratic-equations-using-the-square-root-property

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9.2: Solve Quadratic Equations Using the Square Root Property

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Learning Objectives

By the end of this section, you will be able to:

  • Solve quadratic equations of the form a x 2 = k a x 2 = k using the Square Root Property
  • Solve quadratic equations of the form a ( x – h ) 2 = k a ( x – h ) 2 = k using the Square Root Property

Be Prepared 9.1

Before you get started, take this readiness quiz.

Simplify: 128 . 128 . If you missed this problem, review Example 8.13.

Be Prepared 9.2

Simplify: 32 5 32 5 . If you missed this problem, review Example 8.50.

Be Prepared 9.3

Factor: 9 x 2 − 12 x + 4 9 x 2 − 12 x + 4 . If you missed this problem, review Example 6.23.

A quadratic equation is an equation of the form ax 2 + bx + c = 0, where a ≠ 0 a ≠ 0 . Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax 2 . We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.

Solve Quadratic Equations of the form a x 2 = k a x 2 = k using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x 2 = 9.

We can easily use factoring to find the solutions of similar equations, like x 2 = 16 and x 2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, x = 4 , x = −4 x = 4 , x = −4 and x = 5 , x = −5 . x = 5 , x = −5 .

But what happens when we have an equation like x 2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13) 2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:

If n 2 = m , then n is a square root of m . If n 2 = m , then n is a square root of m .

Since these equations are all of the form x 2 = k , the square root definition tells us the solutions are the two square roots of k . This leads to the Square Root Property .

Square Root Property

If x 2 = k , then

x = k or x = − k or x = ± k . x = k or x = − k or x = ± k .

Notice that the Square Root Property gives two solutions to an equation of the form x 2 = k , the principal square root of k k and its opposite. We could also write the solution as x = ± k . x = ± k . We read this as x equals positive or negative the square root of k .

Now we will solve the equation x 2 = 9 again, this time using the Square Root Property.

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x 2 = 7.

We cannot simplify 7 7 , so we leave the answer as a radical.

Example 9.1

How to solve a quadratic equation of the form ax 2 = k using the square root property.

Solve: x 2 − 50 = 0 . x 2 − 50 = 0 .

Step one is to isolate the quadratic term and make its coefficient one. For the equation x squared minus fifty equals zero, first add fifty to both sides to get x squared by itself. The new equation is x squared equals fifty.

Solve: x 2 − 48 = 0 . x 2 − 48 = 0 .

Solve: y 2 − 27 = 0 . y 2 − 27 = 0 .

The steps to take to use the Square Root Property to solve a quadratic equation are listed here.

Solve a quadratic equation using the square root property.

  • Step 1. Isolate the quadratic term and make its coefficient one.
  • Step 2. Use Square Root Property.
  • Step 3. Simplify the radical.
  • Step 4. Check the solutions.

In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.

Example 9.2

Solve: 3 z 2 = 108 . 3 z 2 = 108 .

Solve: 2 x 2 = 98 . 2 x 2 = 98 .

Solve: 5 m 2 = 80 . 5 m 2 = 80 .

The Square Root Property states ‘If x 2 = k x 2 = k ,’ What will happen if k < 0 ? k < 0 ? This will be the case in the next example.

Example 9.3

Solve: x 2 + 72 = 0 x 2 + 72 = 0 .

Solve: c 2 + 12 = 0 . c 2 + 12 = 0 .

Solve: q 2 + 24 = 0 . q 2 + 24 = 0 .

Our method also works when fractions occur in the equation, we solve as any equation with fractions. In the next example, we first isolate the quadratic term, and then make the coefficient equal to one.

Example 9.4

Solve: 2 3 u 2 + 5 = 17 . 2 3 u 2 + 5 = 17 .

Solve: 1 2 x 2 + 4 = 24 . 1 2 x 2 + 4 = 24 .

Solve: 3 4 y 2 − 3 = 18 . 3 4 y 2 − 3 = 18 .

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator .

Example 9.5

Solve: 2 x 2 − 8 = 41 . 2 x 2 − 8 = 41 .

Solve: 5 r 2 − 2 = 34 . 5 r 2 − 2 = 34 .

Try It 9.10

Solve: 3 t 2 + 6 = 70 . 3 t 2 + 6 = 70 .

Solve Quadratic Equations of the Form a ( x − h ) 2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation of the form a ( x − h ) 2 = k as well. Notice that the quadratic term, x , in the original form ax 2 = k is replaced with ( x − h ).

On the left is the equation a times x square equals k. Replacing x in this equation with the expression x minus h changes the equation. It is now a times the square of x minus h equals k.

The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a , then the Square Root Property can be used on ( x − h ) 2 .

Example 9.6

Solve: 4 ( y − 7 ) 2 = 48 . 4 ( y − 7 ) 2 = 48 .

Try It 9.11

Solve: 3 ( a − 3 ) 2 = 54 . 3 ( a − 3 ) 2 = 54 .

Try It 9.12

Solve: 2 ( b + 2 ) 2 = 80 . 2 ( b + 2 ) 2 = 80 .

Remember when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

Example 9.7

Solve: ( x − 1 3 ) 2 = 5 9 . ( x − 1 3 ) 2 = 5 9 .

Try It 9.13

Solve: ( x − 1 2 ) 2 = 5 4 . ( x − 1 2 ) 2 = 5 4 .

Try It 9.14

Solve: ( y + 3 4 ) 2 = 7 16 . ( y + 3 4 ) 2 = 7 16 .

We will start the solution to the next example by isolating the binomial term.

Example 9.8

Solve: 2 ( x − 2 ) 2 + 3 = 57 . 2 ( x − 2 ) 2 + 3 = 57 .

Try It 9.15

Solve: 5 ( a − 5 ) 2 + 4 = 104 . 5 ( a − 5 ) 2 + 4 = 104 .

Try It 9.16

Solve: 3 ( b + 3 ) 2 − 8 = 88 . 3 ( b + 3 ) 2 − 8 = 88 .

Sometimes the solutions are complex numbers.

Example 9.9

Solve: ( 2 x − 3 ) 2 = −12 . ( 2 x − 3 ) 2 = −12 .

Try It 9.17

Solve: ( 3 r + 4 ) 2 = −8 . ( 3 r + 4 ) 2 = −8 .

Try It 9.18

Solve: ( 2 t − 8 ) 2 = −10 . ( 2 t − 8 ) 2 = −10 .

The left sides of the equations in the next two examples do not seem to be of the form a ( x − h ) 2 . But they are perfect square trinomials, so we will factor to put them in the form we need.

Example 9.10

Solve: 4 n 2 + 4 n + 1 = 16 . 4 n 2 + 4 n + 1 = 16 .

We notice the left side of the equation is a perfect square trinomial. We will factor it first.

Try It 9.19

Solve: 9 m 2 − 12 m + 4 = 25 . 9 m 2 − 12 m + 4 = 25 .

Try It 9.20

Solve: 16 n 2 + 40 n + 25 = 4 . 16 n 2 + 40 n + 25 = 4 .

Access this online resource for additional instruction and practice with using the Square Root Property to solve quadratic equations.

  • Solving Quadratic Equations: The Square Root Property
  • Using the Square Root Property to Solve Quadratic Equations

Section 9.1 Exercises

Practice makes perfect.

Solve Quadratic Equations of the Form ax 2 = k Using the Square Root Property

In the following exercises, solve each equation.

a 2 = 49 a 2 = 49

b 2 = 144 b 2 = 144

r 2 − 24 = 0 r 2 − 24 = 0

t 2 − 75 = 0 t 2 − 75 = 0

u 2 − 300 = 0 u 2 − 300 = 0

v 2 − 80 = 0 v 2 − 80 = 0

4 m 2 = 36 4 m 2 = 36

3 n 2 = 48 3 n 2 = 48

4 3 x 2 = 48 4 3 x 2 = 48

5 3 y 2 = 60 5 3 y 2 = 60

x 2 + 25 = 0 x 2 + 25 = 0

y 2 + 64 = 0 y 2 + 64 = 0

x 2 + 63 = 0 x 2 + 63 = 0

y 2 + 45 = 0 y 2 + 45 = 0

4 3 x 2 + 2 = 110 4 3 x 2 + 2 = 110

2 3 y 2 − 8 = −2 2 3 y 2 − 8 = −2

2 5 a 2 + 3 = 11 2 5 a 2 + 3 = 11

3 2 b 2 − 7 = 41 3 2 b 2 − 7 = 41

7 p 2 + 10 = 26 7 p 2 + 10 = 26

2 q 2 + 5 = 30 2 q 2 + 5 = 30

5 y 2 − 7 = 25 5 y 2 − 7 = 25

3 x 2 − 8 = 46 3 x 2 − 8 = 46

( u − 6 ) 2 = 64 ( u − 6 ) 2 = 64

( v + 10 ) 2 = 121 ( v + 10 ) 2 = 121

( m − 6 ) 2 = 20 ( m − 6 ) 2 = 20

( n + 5 ) 2 = 32 ( n + 5 ) 2 = 32

( r − 1 2 ) 2 = 3 4 ( r − 1 2 ) 2 = 3 4

( x + 1 5 ) 2 = 7 25 ( x + 1 5 ) 2 = 7 25

( y + 2 3 ) 2 = 8 81 ( y + 2 3 ) 2 = 8 81

( t − 5 6 ) 2 = 11 25 ( t − 5 6 ) 2 = 11 25

( a − 7 ) 2 + 5 = 55 ( a − 7 ) 2 + 5 = 55

( b − 1 ) 2 − 9 = 39 ( b − 1 ) 2 − 9 = 39

4 ( x + 3 ) 2 − 5 = 27 4 ( x + 3 ) 2 − 5 = 27

5 ( x + 3 ) 2 − 7 = 68 5 ( x + 3 ) 2 − 7 = 68

( 5 c + 1 ) 2 = −27 ( 5 c + 1 ) 2 = −27

( 8 d − 6 ) 2 = −24 ( 8 d − 6 ) 2 = −24

( 4 x − 3 ) 2 + 11 = −17 ( 4 x − 3 ) 2 + 11 = −17

( 2 y + 1 ) 2 − 5 = −23 ( 2 y + 1 ) 2 − 5 = −23

m 2 − 4 m + 4 = 8 m 2 − 4 m + 4 = 8

n 2 + 8 n + 16 = 27 n 2 + 8 n + 16 = 27

x 2 − 6 x + 9 = 12 x 2 − 6 x + 9 = 12

y 2 + 12 y + 36 = 32 y 2 + 12 y + 36 = 32

25 x 2 − 30 x + 9 = 36 25 x 2 − 30 x + 9 = 36

9 y 2 + 12 y + 4 = 9 9 y 2 + 12 y + 4 = 9

36 x 2 − 24 x + 4 = 81 36 x 2 − 24 x + 4 = 81

64 x 2 + 144 x + 81 = 25 64 x 2 + 144 x + 81 = 25

Mixed Practice

In the following exercises, solve using the Square Root Property.

2 r 2 = 32 2 r 2 = 32

4 t 2 = 16 4 t 2 = 16

( a − 4 ) 2 = 28 ( a − 4 ) 2 = 28

( b + 7 ) 2 = 8 ( b + 7 ) 2 = 8

9 w 2 − 24 w + 16 = 1 9 w 2 − 24 w + 16 = 1

4 z 2 + 4 z + 1 = 49 4 z 2 + 4 z + 1 = 49

a 2 − 18 = 0 a 2 − 18 = 0

b 2 − 108 = 0 b 2 − 108 = 0

( p − 1 3 ) 2 = 7 9 ( p − 1 3 ) 2 = 7 9

( q − 3 5 ) 2 = 3 4 ( q − 3 5 ) 2 = 3 4

m 2 + 12 = 0 m 2 + 12 = 0

n 2 + 48 = 0 . n 2 + 48 = 0 .

u 2 − 14 u + 49 = 72 u 2 − 14 u + 49 = 72

v 2 + 18 v + 81 = 50 v 2 + 18 v + 81 = 50

( m − 4 ) 2 + 3 = 15 ( m − 4 ) 2 + 3 = 15

( n − 7 ) 2 − 8 = 64 ( n − 7 ) 2 − 8 = 64

( x + 5 ) 2 = 4 ( x + 5 ) 2 = 4

( y − 4 ) 2 = 64 ( y − 4 ) 2 = 64

6 c 2 + 4 = 29 6 c 2 + 4 = 29

2 d 2 − 4 = 77 2 d 2 − 4 = 77

( x − 6 ) 2 + 7 = 3 ( x − 6 ) 2 + 7 = 3

( y − 4 ) 2 + 10 = 9 ( y − 4 ) 2 + 10 = 9

Writing Exercises

In your own words, explain the Square Root Property.

In your own words, explain how to use the Square Root Property to solve the quadratic equation ( x + 2 ) 2 = 16 ( x + 2 ) 2 = 16 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations of the form a times x squared equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Solving Quadratic Equations By Taking Square Roots Worksheets

Solving quadratic equations by taking square roots worksheets gives kids an insight into how to attempt questions that required them to use the concept of square roots to find the value of an unknown variable.

Benefits of Solving Quadratic Equations by Taking Square Roots Worksheets

One of the major advantages of solving quadratic equations by taking square roots worksheet is that the questions are arranged in increasing order of difficulty. Kids can first build confidence in solving more straightforward sections and then attempt challenging questions.

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Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

  • Solve Quadratic Equations by Factoring
  • Solve Quadratic Equations by Completing the Square
  • Quadratic Formula Worksheet (real solutions)
  • Quadratic Formula Worksheet (complex solutions)
  • Quadratic Formula Worksheet (both real and complex solutions)
  • Discriminant Worksheet
  • Sum and Product of Roots
  • Radical Equations Worksheet

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Solving Quadratic Equations Using Extracting Square Roots

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  1. Solving Quadratic Equations by Completing the Square

  2. Solving Quadratic Equations Using Completing The Square

  3. Q 3 Solving Quadratic Equations Using Square Roots

  4. 9.3 Solving Quadratic Equations Using Square Roots

  5. Solving Quadratic Equations using Square Roots

  6. 9.3 Solving Quadratic Equations Using Square Roots

COMMENTS

  1. PDF Solving Quadratic Roots

    Solving Quadratic Equations with Square Roots Date_____ Period____ Solve each equation by taking square roots. 1) k2 = 76 {8.717 , −8.717} 2) k2 = 16 {4, −4} 3) x2 = 21 ... Create your own worksheets like this one with Infinite Algebra 1. Free trial available at KutaSoftware.com. Title: Solving Quadratic Roots

  2. Solving Quadratic Equations by Taking Square Roots Worksheets

    Solve Quadratic Equations by Taking Square Roots - Level 1. Push-start your practice of finding the real and complex roots of quadratic equations with this set of pdf worksheets presenting 30 pure quadratic equations. Note that the coefficient of the leading term is 1 in every equation. Hence, simply rewrite the given equation in the form of x ...

  3. PDF 4.3 Solving Quadratic Equations Using Square Roots

    Section 4.3 Solving Quadratic Equations Using Square Roots 211 Solving a Quadratic Equation Using Square Roots Solve (x − 1)2 = 25 using square roots.SOLUTION (x − 1)2 = 25 Write the equation.x − 1 = ±5 Take the square root of each side. x = 1 ± 5 Add 1 to each side. So, the solutions are x = 1 + 5 = 6 and x = 1 − 5 = −4. Check Use a graphing calculator to check

  4. Solving Quadratic Equations by Square Root Method

    The solutions to this quadratic formula are [latex]x = 3 [/latex] and [latex]x = - \,3 [/latex]. Example 4: Solve the quadratic equation below using the Square Root Method. The two parentheses should not bother you at all. The fact remains that all variables come in the squared form, which is what we want. This problem is perfectly solvable ...

  5. 9.2: Solve Quadratic Equations Using the Square Root Property

    Put the equation in standard form. x2 − 9 = 0. Factor the difference of squares. (x − 3)(x + 3) = 0. Use the Zero Produce Property. x − 3 = 0 x − 3 = 0. Solve each equation. x = 3 x = − 3. We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares.

  6. PDF Title: Solving Quadratic Equations using Square Roots Math 107

    Purpose: This is intended to refresh your knowledge about solving quadratic equations using square roots. ≠ 0 . For example, 3 x. 2 x = 2 x , and x ( x + 6) = 14 are all quadratic equations. Note that the second two equations would require a couple algebraic steps to be put into the form shown above. 2 x − 9 = 0 by factoring; ( x − 3)( x ...

  7. PDF Solving Quadratic Equations by Square Roots

    IMC Practice Worksheet Name: _____ Date: _____ Solving Quadratic Equations by Square Roots Solve the equation by square roots. Remember 1. x2 = 324 22 2. x - 81 = 0 3. 5x 180 = 0 4. 23x - 100 = 332 2 5. ⅔ x2 - 8 = 16 6. ½ x - 5 = 5

  8. 10.1 Solve Quadratic Equations Using the Square Root Property

    In this chapter, we will use three other methods to solve quadratic equations. Solve Quadratic Equations of the Form ax 2 = k Using the Square Root Property. We have already solved some quadratic equations by factoring. Let's review how we used factoring to solve the quadratic equation x 2 = 9 x 2 = 9.

  9. Algebra 1

    Include Quadratic Functions Worksheet Answer Page. Now you are ready to create your Quadratic Functions Worksheet by pressing the Create Button. If You Experience Display Problems with Your Math Worksheet. This Algebra 1 - Quadratic Functions Worksheet produces problems for solving quadratic equations by taking the square root.

  10. Solving quadratics by taking square roots

    For example, to solve the equation 2 x 2 + 3 = 131 we should first isolate x 2 . We do this exactly as we would isolate the x term in a linear equation. 2 x 2 + 3 = 131 2 x 2 = 128 Subtract 3. x 2 = 64 Divide by 2. x 2 = 64 Take the square root. x = ± 8. Now solve a few similar equations on your own. Problem 7.

  11. 9.1 Solve Quadratic Equations Using the Square Root Property

    Solve Quadratic Equations of the Form a(x − h) 2 = k Using the Square Root Property. We can use the Square Root Property to solve an equation of the form a(x − h) 2 = k as well. Notice that the quadratic term, x, in the original form ax 2 = k is replaced with (x − h). The first step, like before, is to isolate the term that has the variable squared.

  12. Solving Quadratic Equations

    Solving Quadratic Equations - Worksheet Solve each equation by using the square root property. 1. x2 = 49 2. x2 11 = 0 3. y2 = 20 4. (y 3)2 = 4 5. (4x+9)2 = 6

  13. Solve quadratic equations by taking square roots (practice)

    Quadratics by taking square roots. Google Classroom. You might need: Calculator. Solve for x . Enter the solutions from least to greatest. ( x + 5) 2 − 64 = 0. lesser x =. greater x =. Show Calculator.

  14. 9.2: Solve Quadratic Equations Using the Square Root Property

    Solve Quadratic Equations of the Form a(x − h) 2 = k Using the Square Root Property. We can use the Square Root Property to solve an equation of the form a(x − h) 2 = k as well. Notice that the quadratic term, x, in the original form ax 2 = k is replaced with (x − h). The first step, like before, is to isolate the term that has the variable squared.

  15. Solving quadratics by completing the square

    Let's start with the solution and then review it more closely. ( 1) x 2 + 6 x = − 2 ( 2) x 2 + 6 x + 9 = 7 Add 9, completing the square. ( 3) ( x + 3) 2 = 7 Factor the expression on the left. ( 4) ( x + 3) 2 = ± 7 Take the square root. ( 5) x + 3 = ± 7 ( 6) x = ± 7 − 3 Subtract 3. In conclusion, the solutions are x = 7 − 3 and x = − ...

  16. Solving Quadratic Equations By Taking Square Roots Worksheets

    Benefits of Solving Quadratic Equations by Taking Square Roots Worksheets. One of the major advantages of solving quadratic equations by taking square roots worksheet is that the questions are arranged in increasing order of difficulty. Kids can first build confidence in solving more straightforward sections and then attempt challenging questions.

  17. Quadratic Equation Worksheets (pdfs)

    Enjoy these free sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key. Solve Quadratic Equations by Factoring. Solve Quadratic Equations by Completing the Square. Quadratic Formula Worksheets.

  18. IXL

    IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more. 0.

  19. Solving Quadratic Equations by Square Roots worksheet

    30/12/2021. Country code: US. Country: United States. School subject: Math (1061955) Main content: Quadratic Equations (2037465) solve quadratic equations by square roots. Loading ad...

  20. Solving Quadratic Equations Using Extracting Square Roots worksheet

    ID: 1089331. 15/06/2021. Country code: PH. Country: Philippines. School subject: Math (1061955) Main content: Solving Quadratic Equations (1900566) Evaluation for Grade 9 Students. Other contents: Extracting Square Roots.