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Quadratic Equation Worksheets (pdfs)

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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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Quadratic equation is one of the most important and potentially high-scoring topics asked in different banking and insurance exams, such as IBPS PO, SBI PO, SBI Clerk, IBPS Clerk, RRB Assistant, RRB Scale 1, LIC Assistant, LIC AAO, etc. You can expect a set of 5 questions  on this topic in the prelims of every banking and insurance examination. Achieving proficiency in this topic demands keen observational skills. Nonetheless, with dedicated practice, you can attain mastery and achieve a perfect score in this area. Smartkeeda offers a diverse range of Quadratic Equation questions with solutions to facilitate effective practice and enhance your prospects of achieving a high score.  

Identifying Quadratic Patterns Question

A quadratic equation, denoted by the variable x, takes the form of ax 2  + bx + c = 0, where a, b, and c represent real numbers, with a ≠ 0. For instance, 2x 2  + x - 300 = 0 is a quadratic equation. However, to establish the standard representation of this equation, we arrange the terms of p(x) in descending order of their degrees, resulting in ax 2  + bx + c = 0, with a ≠ 0. This particular form, ax 2  + bx + c = 0, where 'a' is not equal to zero, is referred to as the standard form of a quadratic equation.  

Understanding the Quadratic Formula

Also known as the Sridharacharya formula , the quadratic formula is a formula that provides the two solutions to a quadratic equation. The Quadratic formula stands as the most straightforward method for determining the roots of a quadratic equation. In cases where certain quadratic equations resist easy factorization, the Quadratic formula offers a convenient and efficient means to swiftly calculate the roots.  

The Quadratic Formula

Solve using Quadratic formula 2x 2  - 7x + 3 = 0 Solution: Comparing the equation with the general form ax 2  + bx + c = 0 gives, a = 2, b = -7, and c = 3 Now, calculate the discriminant (b 2  - 4ac): b 2  - 4ac = (-7) 2  - 4 * 2 * 3 = 49 - 24 = 25 Now, substitute the values into the quadratic formula: x1 = (-b + √(b 2  - 4ac)) / (2a) x1 = (-(-7) + √25) / (2 * 2) x1 = (7 + 5) / 4 = 3 x2 = (-b - √(b 2  - 4ac)) / (2a) x2 = (-(-7) - √25) / (2 * 2) x2 = (7 - 5) / 4 = 1/2 So, the roots of the equation 2x 2 - 7x + 3 = 0 are x1 = 3 and x2 = 1/2.  

Factoring Quadratic Equations

Factoring quadratics is a technique used to represent the quadratic equation ax^2 + bx + c = 0 as a multiplication of its linear factors in the form (x - p) (x - q), where p and q represent the roots of the quadratic equation ax^2 + bx + c = 0. Factoring Techniques Quadratic equations can be factorized through various methods such as

  • Splitting the middle term,
  • Using quadratic formula or Shridharacharya formula
  • Completing the square or square root method Understanding the pattern of Quadratic Equations asked in bank exams

Understanding the pattern of Quadratic Equations asked in bank exams

Bank exams often include these types of Quadratic Equation questions to assess candidates' problem-solving skills and their ability to discern relationships between variables, making it a critical component of the quantitative section. Questions on Quadratic Equations are asked in the form of inequalities in the Quantitative Aptitude section. Generally, two quadratic equations in two different variables are given. You have to solve both of the Quadratic equations to get to know the relation between both variables. Suppose we have two variables ‘x’ and ‘y’. The relationship between the variables can be any one of the following:   x > y x < y x = y or relation can’t be established between x & y x ≥ y x ≤ y

Practice Exercises

I.   x2 – 25x + 114 = 0 II.  y2 – 10y + 24 = 0

  • if x > y
  • if x < y
  • if x = y or relationship between x and y can't be established

Tips for Efficient Quadratic Equation Solving

When solving quadratic equations, it's important to keep the following points in mind to ensure accurate and efficient problem-solving:

  • Recognize that a quadratic equation is in the form ax^2 + bx + c = 0
  • After finding potential solutions, ensure they satisfy the original equation.
  • Carefully handle the signs (+/-) in the quadratic formula to avoid calculation errors.
  • Observe carefully while comparing the roots of given equations

Frequently Asked Questions

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  • Quadratic Equation Questions

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Basic Quadratic Equation Questions

Quadratic equations are an important part of algebra, and as students, we must all be familiar with their definition and the ways of solving quadratic equation problems. In this article, we are going to familiarize the students with all the concepts surrounding quadratic equations and the methods of solving problems related to this topic. A short definition of a quadratic equation would be: a quadratic equation is a second-degree polynomial, which we represent as ‘ax 2 + bx + c’ in general.

In this representation, a cannot be equal to 0 and b,c are known as coefficients and are constant by nature. With this basic introduction, let's move forward with a formal definition, formulae and detailed solutions to quadratic equation questions to enable better understanding.

Why are Quadratic Equations Important?

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and even computer science. They help us model real-world scenarios like projectile motion, population growth, and electrical circuit analysis. Understanding quadratics is crucial for success in higher-level math and science courses.

Definition of Quadratic Equations

A quadratic equation is a polynomial where the highest power of the variable is 2. We generally represent it as ax 2 + bx + c. Here a, b and c are real numbers or constants, and x is the variable. In this case, the value of a cannot be 0 as that would remove the x 2 term, and the equation won't be quadratic after that.

A quadratic equation is an equation of second degree with more than two terms. It means that at least one of the terms of the equation is squared. In the above-given equation. The answer to the equation also known as the roots of the equation is the value of the “x”. The value of the “x” has to satisfy the equation. 

Some examples of quadratic equations can be as follows:

56x 2 + ⅔ x + 1, where a = 56, b = ⅔ and c = 1.

-4/3 x 2 + 64x - 30, where a = -4/3, b = 64 and c = -30.

Roots of a Quadratic Equation

To solve basic quadratic equation questions or any quadratic equation problems, we need to solve the equation. Solving quadratic equations gives us the roots of the polynomial. The roots of the equation are the values of x at which ax 2 + bx + c = 0. Since a quadratic equation is a polynomial of degree 2, we obtain two roots in this case.

There are several methods for solving quadratic equation problems, as we can see below:

Factorization Method.

Completing The Square Method.

Quadratic Equation Formula.

Quadratic Equation Formula

So what is the quadratic equation formula? The quadratic equation formula or the Sridharacharya Formula is a method for finding out the roots of two-degree polynomials. This formula helps solve quadratic equation problems. The formula is as given below:

\[ x = \frac{-b \pm\sqrt{b^{2}-4ac}}{2a} \]

Where x represents the roots of the equation and (b 2 −4ac) is the discriminant.

By finding out the value of the discriminant, we can predict the nature of the roots. There are three possibilities with three different implications:

Two distinct roots which are real, if b 2 - 4ac > 0.

Two real roots equal in magnitude, if b 2 - 4ac = 0.

Imaginary roots or absence of real roots if b 2 - 4ac < 0.

Now that the basic principles of quadratic equations are clear, we will move on to some solved examples. But before that, let us list some quadratic equation questions for the students to solve.

Quadratic Equation Practice Questions

The following are a list of questions for you to solve once you have gone through the quadratic equation questions and answers in the solved examples section:

Find the determinant of the following quadratic equations: 2x 2 + 3x + 6, 70x 2 + 49 + 14, ⅔ y 2 + 63y + 42.

Find the roots of the following quadratic equations: x 2 - 45x + 324, 2x 2 - 22x + 42, ½ x 2 + 2x + 4.

The product of two consecutive numbers is 420, and their sum is 41. Find the numbers.

Before solving these, let's check out the solved examples with questions and answers on the quadratic equation.

Solved Examples

1. Solve x 2 + 5x + 6 = 20 by factorization method.

Solution: The given polynomial or quadratic equation is

x 2 + 5x + 6 = 20

Solving by factorization method, 

x 2 + 5x + 6 - 20 = 0.

or, x 2 + 5x - 14 = 0

or, x 2 - 2x + 7x - 14 = 0

or, x(x - 2) +7(x - 2) = 0

or, (x - 2)(x + 7) = 0

or, (x - 2) = 0, (x + 7) = 0 

or, x = +2, -7.

2. Solve 2x 2 - 5x + 3 using the quadratic equation formula.

Solution: The quadratic equation formula is:

\[ x = \frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \]

The determinant or b 2 -4ac = (-5) 2 - 4 × 3 × 2 = 25 - 24 = 1

\[ \sqrt{b^{2} - 4ac} = 1 \]

Therefore, \[ x= \frac{-(-5)\pm1}{2\times2} \]

\[ x = \frac{5+1}{4} = \frac{6}{4} = \frac{3}{2} \]

\[ x = \frac{5-1}{4} = \frac{4}{4} = 1 \]

Thus the roots of the equation are 3/2 and 1.

3. Solve the quadratic equation: \[x^2 - 4x + 4 = 0\]

Solution: Given quadratic equation is a perfect square trinomial, so we can factor it directly.

\[(x - 2)^2 = 0\]

Taking the square root of both sides:

\[x - 2 = 0\]

Solving for \(x\):

4. Solve the quadratic equation: \[2x^2 - 5x + 2 = 0\]

Solution: Using the quadratic formula:

\[x = \dfrac{5 \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}\]

\[x = \dfrac{5 \pm \sqrt{25 - 16}}{4}\]

\[x = \dfrac{5 \pm \sqrt{9}}{4}\]

\[x = \dfrac{5 \pm 3}{4}\]

Two solutions:

\[x = \dfrac{8}{4} = 2\]

\[x = \dfrac{2}{4} = \dfrac{1}{2}\]

5. Solve the quadratic equation: \[3x^2 + 7x + 2 = 0\]

\[x = \dfrac{-7 \pm \sqrt{7^2 - 4(3)(2)}}{2(3)}\]

\[x = \dfrac{-7 \pm \sqrt{49 - 24}}{6}\]

\[x = \dfrac{-7 \pm \sqrt{25}}{6}\]

\[x = \dfrac{-7 \pm 5}{6}\]

\[x = \dfrac{-2}{6} = -\dfrac{1}{3}\]

\[x = \dfrac{-12}{6} = -2\]

6. Solve the quadratic equation: \[4x^2 + 8x + 2 = 0\]

Solution: Divide the equation by the common factor (in this case, 2):

\[2x^2 + 4x + 1 = 0\]

Using the quadratic formula:

\[x = \dfrac{-4 \pm \sqrt{4^2 - 4(2)(1)}}{2(2)}\]

\[x = \dfrac{-4 \pm \sqrt{16 - 8}}{4}\]

\[x = \dfrac{-4 \pm \sqrt{8}}{4}\]

\[x = \dfrac{-4 \pm 2\sqrt{2}}{4}\]

\[x = \dfrac{-2 \pm \sqrt{2}}{2}\]

\[x = \dfrac{-2 + \sqrt{2}}{2}\]

\[x = \dfrac{-2 - \sqrt{2}}{2}\]

7. Solve the quadratic equation by factorization method: \[x^2 - 5x + 6 = 0\]

Solution: Factor the quadratic equation:

\[(x - 2)(x - 3) = 0\]

Setting each factor to zero:

\[x - 2 = 0 \quad \text{or} \quad x - 3 = 0\]

\[x = 2 \quad \text{or} \quad x = 3\]

8. Solve the quadratic equation: \[6x^2 - 11x + 4 = 0\]

\[x = \dfrac{11 \pm \sqrt{(-11)^2 - 4(6)(4)}}{2(6)}\]

\[x = \dfrac{11 \pm \sqrt{121 - 96}}{12}\]

\[x = \dfrac{11 \pm \sqrt{25}}{12}\]

\[x = \dfrac{11 \pm 5}{12}\]

\[x = \dfrac{16}{12} = \dfrac{4}{3}\]

\[x = \dfrac{6}{12} = \dfrac{1}{2}\]

9. Solve the quadratic equation: \[2x^2 + 3x - 5 = 0\]

\[x = \dfrac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)}\]

\[x = \dfrac{-3 \pm \sqrt{9 + 40}}{4}\]

\[x = \dfrac{-3 \pm \sqrt{49}}{4}\]

\[x = \dfrac{-3 \pm 7}{4}\]

\[x = \dfrac{4}{4} = 1\]

\[x = \dfrac{-10}{4} = -\dfrac{5}{2}\]

10. Solve the quadratic equation: \[5x^2 - 4x - 3 = 0\]

\[x = \dfrac{4 \pm \sqrt{(-4)^2 - 4(5)(-3)}}{2(5)}\]

\[x = \dfrac{4 \pm \sqrt{16 + 60}}{10}\]

\[x = \dfrac{4 \pm \sqrt{76}}{10}\]

\[x = \dfrac{4 \pm 2\sqrt{19}}{10}\]

\[x = \dfrac{2 \pm \sqrt{19}}{5}\]

\[x = \dfrac{2 + \sqrt{19}}{5}\]

\[x = \dfrac{2 - \sqrt{19}}{5}\]

This is all about the roots of quadratic equations and their formulas. Learn the formulas and find out how they are used to derive the roots of an equation easily. 

By diving into this chapter, you'll build a solid understanding of quadratic equations. Mastering these equations will empower you to effortlessly solve any quadratic problem, use your skills in real-life situations, and amaze your teachers with your math abilities. Keep in mind that practice is crucial! Take on the practice questions and reach out for assistance if necessary. The realm of math is ready for you, and quadratic equations are your ticket to exploring its intriguing complexities!

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FAQs on Quadratic Equation Questions

1. Define a quadratic equation along with suitable examples. also, state the quadratic equation formula.

A quadratic equation is a polynomial where the highest power of the variable is neither more nor less than 2. So essentially, a quadratic equation is a polynomial of degree 2. We represent such an equation in a general format as ax 2 + bx + c, where a, b and c are known as the coefficients or the constants of the equation. The thumb rule for quadratic equations is that the value of a cannot be 0. The x in the expression is the variable. This algebraic expression, when solved, will yield two roots.

Some examples of quadratic equations are:

3x 2 + 4x + 7 = 34

x 2 + 8x + 12 = 40

The quadratic equation formula is a method for solving quadratic equation questions. The formula is as follows:

where x represents the roots of the equation.

2. What are the roots of a quadratic equation? What are the zeroes of a polynomial?  

The roots of a quadratic equation are the values obtained when we solve the equation. They are those values of x for which the expression ax 2 +bx+c becomes equal to 0. These values are also known as the zeroes of the polynomial. Since a quadratic equation is essentially a polynomial of degree 2, we get two roots after solving the given polynomial.

As we practice more and more quadratic equation sums, our ideas regarding which method to use while solving a given question will get clearer.

3. How can I solve a quadratic equation?

Quadratic equation is an equation with more than one term in it and at least one of the terms having degree 2. Its general form is ax 2  + bx + c, whereas a,b,c are real numbers and a is not equal to zero. The values which satisfy the “x” in the equation are the solution for the quadratic equation. They are also known as the roots of the equation. The quadratic equation can be solved in the algebraic method and graphical method.

In the algebraic method, the equation is reduced to the roots by shifting terms from L.H.S to the R.H.S and using different mathematical operations. In the graphic method, the equation is solved by drawing it on the map and solving it using the parabola the equation makes on the graph. The value of “a” determines whether the graph of the equation is concave parabola or convex parabola. The value of the discriminant decides whether the curve will intersect the x-axis or not.

4.  What are the applications of quadratic equations?

A quadratic equation is a polynomial equation with more than one term and at least one of the terms having 2 as square. It is generally used in different situations in day-to-day life. In constructing rooms and boxes of different geometric shapes. If you want to construct a box made of wood with 5 square feet dimensions, you can write a quadratic equation to measure its area and calculate the material required. 

It can also be used in selling something and calculating the profit and loss you may incur after selling the good. To know it, you can simply form a quadratic equation. It can also be used in athletics while throwing objects like a javelin, shot put ball, etc. It can also be used to calculate the distance. Generally, when someone travels up and down the river uses this equation to measure the distance to be travelled. 

5. How are quadratic equations used in athletics and construction?

Quadratic equations are equations with at least one term having 2 as a square and it has more than one term in the equation preferably, four terms. The general form of the equation is ax 2  + bx+c whereas a,b,c are real numbers and a is not equal to zero. The values which satisfy the “x” in the equation are the solution for the quadratic equation. They are also known as the roots of the equation. In exams, they have preferable weightage and practising them correctly will improve the marks. On the other hand, they are also used in real-life situations. In athletics, it is used to measure the speed and force to be applied to throw an object like an arrow, shot put the ball, discus, etc. They use the velocity equation to measure the height of the ball from which it should be thrown. In the field of construction, a quadratic equation is framed with the known dimensions of the building or a room to figure out unknown values like the area, perimeter, etc.

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Quadratic Equation Questions

Quadratic equation questions are provided here for Class 10 students. A quadratic equation is a second-degree polynomial which is represented as ax 2 + bx + c = 0, where a is not equal to 0. Here, a, b and c are constants, also called coefficients and x is an unknown variable. Also, learn Quadratic Formula here.

Solving the problems based on quadratics will help students to understand the concept very well and also to score good marks in this section. All the questions are solved here step by step with a detailed explanation. In this article, we will give the definition and important formula for solving problems based on quadratic equations. The questions given here is in reference to the CBSE syllabus and NCERT curriculum. 

Definition of Quadratic Equation

Usually, the quadratic equation is represented in the form of ax 2 +bx+c=0, where x is the variable and a,b,c are the real numbers & a ≠ 0. Here, a and b are the coefficients of x 2 and x, respectively. So, basically, a quadratic equation is a polynomial whose highest degree is 2. Let us see some examples:

  • 3x 2 +x+1, where a=3, b=1, c=1
  • 9x 2 -11x+5, where a=9, b=-11, c=5

Roots of Quadratic Equations:

If we solve any quadratic equation, then the value we obtain are called the roots of the equation. Since the degree of the quadratic equation is two, therefore we get here two solutions and hence two roots.

There are different methods to find the roots of quadratic equation, such as:

  • Factorisation
  • Completing the square
  • Using quadratic formula

Learn: Factorization of Quadratic equations

Quadratic Equation Formula:

The quadratic formula to find the roots of the quadratic equation is given by:

\(\begin{array}{l}x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\end{array} \)

Where b 2 -4ac is called the discriminant of the equation.

Based on the discriminant value, there are three possible conditions, which defines the nature of roots as follows:

  • two distinct real roots, if b 2 – 4ac > 0
  • two equal real roots, if b 2 – 4ac = 0
  • no real roots, if b 2 – 4ac < 0

Also, learn quadratic equations for class 10 here.

Quadratic Equations Problems and Solutions

1. Rahul and Rohan have 45 marbles together. After losing 5 marbles each, the product of the number of marbles they both have now is 124. How to find out how many marbles they had to start with.

Solution: Say, the number of marbles Rahul had be x.

Then the number of marbles Rohan had = 45 – x.

The number of marbles left with Rahul after losing 5 marbles = x – 5

The number of marbles left with Rohan after losing 5 marbles = 45 – x – 5 = 40 – x

The product of number of marbles = 124

(x – 5) (40 – x) = 124

40x – x 2 – 200 + 5x = 124

– x 2 + 45x – 200 = 124

x 2 – 45x + 324 = 0

This represents the quadratic equation. Hence by solving the given equation for x, we get;

x = 36 and x = 9

So, the number of marbles Rahul had is 36 and Rohan had is 9 or vice versa.

2. Check if x(x + 1) + 8 = (x + 2) (x – 2) is in the form of quadratic equation.

Solution: Given,

x(x + 1) + 8 = (x + 2) (x – 2)

Cancel x 2 both the sides.

Since, this expression is not in the form of ax 2 +bx+c, hence it is not a quadratic equation.

3. Find the roots of the equation 2x 2 – 5x + 3 = 0 using factorisation.

2x 2 – 5x + 3 = 0

2x 2 – 2x-3x+3 = 0

2x(x-1)-3(x-1) = 0

(2x-3) (x-1) = 0

2x-3 = 0; x = 3/2

(x-1) = 0; x=1

Therefore, 3/2 and 1 are the roots of the given equation.

4. Solve the quadratic equation 2x 2 + x – 300 = 0 using factorisation.

Solution: 2x 2 + x – 300 = 0

2x 2 – 24x + 25x – 300 = 0

2x (x – 12) + 25 (x – 12) = 0

(x – 12)(2x + 25) = 0

x-12=0; x=12

(2x+25) = 0; x=-25/2 = -12.5

Therefore, 12 and -12.5 are two roots of the given equation.

Also, read   Factorisation .

5. Solve the equation x 2 +4x-5=0.

x 2 + 4x – 5 = 0

x 2 -1x+5x-5 = 0

x(x-1)+5(x-1) =0

(x-1)(x+5) =0

Hence, (x-1) =0, and (x+5) =0

similarly, x+5 = 0

x=-5 & x=1

6. Solve the quadratic equation 2x 2 + x – 528 = 0, using quadratic formula.

Solution: If we compare it with standard equation, ax 2 +bx+c = 0

a=2, b=1 and c=-528

Hence, by using the quadratic formula:

Now putting the values of a,b and c.

x=64/4 or x=-66/4

x=16 or x=-33/2

7. Find the roots of x 2 + 4x + 5 = 0, if any exist, using quadratic formula.

Solution: To check whether there are real roots available for the quadratic equation, we need the find the discriminant value.

D = b 2 -4ac = 4 2 – 4(1)(5) = 16-20 = -4

Since the square root of -4 will not give a real number. Hence there is no real roots for the given equation.

8. Find the discriminant of the equation: 3x 2 -2x+⅓ = 0.

Solution: Here, a = 3, b=-2 and c=⅓

Hence, discriminant, D = b 2 – 4ac

D = (-2) 2 -4(3)(⅓)

Video Lesson

Quadratic equation worksheet.

quadratic equation solved questions pdf

Practice Questions

Solve these quadratic equations and find the roots. 

  • x 2 -5x-14=0         [Answer: x=-2 & x=7]
  • X 2  = 11x -28       [Answer: x=4 & x = 7]
  • 6x 2 – x = 5            [Answer: x=-⅚ & x = 1]
  • 12x 2 = 25x          [Answer: x=0 & x=25/12]

Frequently Asked Questions on Quadratic Equations

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Important Quadratic Equation Qs With detailed Solutions | Download Now

Quadratic Equation Questions

Quadratic Equation Questions: Quadratic Equation Questions are very important for every Banking, Insurance, SSC, Railways, and other government exams. This topic requires a lot of practice and the right approach to score good marks in this section. To help you with your practice we have brought you a free ebook on difficult Quadratic Equation Questions with detailed solutions For practice. Download the free ebook now and practice as much as you can to ace this section.

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Here’s a sneek peek into this ebook.

Q1. In the following questions, two equations numbered I and II are given. You have to solve both equations and choose the correct option.

I. 10x 2 -11x +3 = 0 II. y 2 +30y +224 = 0

  • x = y or relationship cannot be established

Answer key: 2

Q2. In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option.

I. 10x 2 -11x -35 = 0 II. y 2 -19y -92 = 0

Answer key: 5

Q3. In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option.

I. 10x 2 -17x -20 = 0 II. y 2 +22y +40 = 0

Q4. In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option.

I. 10x 2 -19x -15 = 0 II. y 2 -12y -189 = 0

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This was all from us in this ebook, Quadratic Equation Questions. Download the free ebook now and start practicing for different upcoming exams. Also, stay tuned with Oliveboard for different parts for Quadratic Equation Ebook.

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  • Exporting Data from scripts in R Programming
  • Calculate Hyperbolic cosine of a value in R Programming - cosh() Function
  • Calculate sine of a value in R Programming - sin() Function
  • Calculate Hyperbolic sine of a value in R Programming - sinh() Function
  • Calculate Hyperbolic tangent of a value in R Programming - tanh() Function
  • Calculate tangent of a value in R Programming - tan() Function
  • Calculate Inverse sine of a value in R Programming - asin() Function
  • Calculate Square root of a number in R Language - sqrt() Function
  • Calculate Inverse cosine of a value in R Programming - acos() Function
  • Calculate Inverse tangent of a value in R Programming - atan() Function
  • Convert Degree value to Radian value in R Programming - deg2rad() Function
  • Convert Radian value to Degree value in R Programming - rad2deg() Function
  • Rounding off to the ceiling or floor value in R Programming - trunc() Function
  • Rounding off a value to specific digits in R Programming - round() Function
  • Calculate exponential of a number in R Programming - exp() Function
  • Calculate cosine of a value in R Programming - cos() Function

Solve Quadratic Equation in R

R language is the language of data visualization and data analytics. It is used to solve complex problems or to visualize the given datasets. In this article, we are going to learn how we can solve quadratic equations using R Programming Language .

What is a Quadratic Formula?

The quadratic formula is used to find the x-intercepts of the quadratic equation, We use the quadratic formula to solve the quadratic equation.

[Tex]x = \frac{{-b \pm \sqrt{{b^2 – 4ac}}}}{{2a}} [/Tex]

Where ‘x’ is given or we to find out and ‘a’, ‘b’, and ‘c’ are known numbers such that a!=0

  • If a=0, then the equation is linear and not a quadratic equation.
  • a,b,c are the coefficients of the equation, they are also called linear coefficients, quadratic coefficients, or free term.

Understanding Discriminant and its Three Cases

Notice the above formula and the square root part of it. ( b 2 -4ac ) is called the discriminant and it is used to determine the real roots that we will get or not.

  • Case 1: (b 2 -4ac) > 0 If this case is true, then we would have two distinct solutions or two real roots.
  • Case 2: (b2-4ac) = 0 If this case is true, then There would be one solution or one real root for the equation.
  • Case 3: (b2-4ac) < 0 If this case is true, then there would be no real roots and hence we don’t have to solve the equation at all.

These three cases are used to determine whether we get the solution or not, if the discriminant is negative we don’t solve the quadratic equation.

Solving Quadratic using R language

Now we are going to write a code using R to solve quadratic equations using quadratic formulas.

Initially, we created a function named ‘ quadRoots()’ in which we define variable and calculate the discriminant. Based on discriminant value, the function call the correct if-else ladder.

  • The user has to pass three arguments in the function to get the result. The parameters represent ‘a’, ‘b’, ‘c’.
  • The ‘nsmall’ argument is passed in ‘format()’ to set the precision of a number of digits right to the decimal point.

[1] "You have chosen the quadratic equation 2x^2 + 43x + 45." [1] "The two x-intercepts for the quadratic equation are -1.10311 and -20.39689."

Here, we get (b 2 -4ac) > 0, that’s why we got two solutions.

  • We stored the discriminant value in x_int_plus and x_int_neg . As we have to calculate the discriminant, once taking the ‘ +’ sign and once taking the ‘-‘ sign.

[1] "You have chosen the quadratic equation 2x^2 + 3x + 4." [1] "This quadratic equation has no real numbered roots."

After writing the function in R call the function with any three parameters as a, b, and c.

  • It calculates the discriminant (b 2 – 4ac) to execute the correct if-else ladder.
  • If the discriminant is negative, the function returns a message indicating that the quadratic equation has no real-numbered roots.
  • The function calculates and returns the two distinct real roots using the quadratic formula if the discriminant is positive.
  • If it is zero, then the equation has only one real root.

[1] "You have chosen the quadratic equation 2x^2 + 4x + 2." [1] "The quadratic equation has only one root. This root is -1"

Here we get ‘D=0’ , which means that the quadratic equation has only one real root.

  • Calculating the roots of quadratic equations that cant be solved by using a simple factoring technique.

Solving Quadratic using another function in R language

We have created a function with the name ‘result()’ that will take three arguments representing ‘a’, ‘b’, ‘c’.

  • Then, we have a second function with the name ‘discriminant()’ that will help us to find the discriminant.
  • The logic here is simple we follow the quadratic formula for solving the equation that can’t be solved by using the factoring technique.

[1] -0.5763735 -6.2765677

Here, D>0 that’s why we got two real roots.

[1] "There are no real roots."

Here, D<0 that’s why it doesn’t have any real roots.

Here, D=0 that’s why the equation has only one root.

R is useful for solving mathematical problems not simple but also complex problems. This language is majorly used for data visualization and solving complex mathematical problems. In the article, we have learned to solve quadratic equations using quadratic formulas.

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    Output: [1] "You have chosen the quadratic equation 2x^2 + 3x + 4." [1] "This quadratic equation has no real numbered roots." After writing the function in R call the function with any three parameters as a, b, and c. . It calculates the discriminant (b 2 - 4ac) to execute the correct if-else ladder.; If the discriminant is negative, the function returns a message indicating that the ...