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Algebra concepts that pupils can work on here include:

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- 1. Multiple-choice 30 seconds 1 pt What is the first step in solving this equation: 5x + 7 = 12 Divide by 5 Subtract 5 Multiply by 7 Subtract 7
- 2. Multiple-choice 30 seconds 1 pt What is the first step to complete when solving this equation? 10 - 3y = 4 Add 10 Subtract 10 Add 3 Divide by -3
- 3. Multiple-choice 30 seconds 1 pt What is the first step to complete when solving this equation? 3y - 10 = 4 Add 10 Subtract 10 Add 3 Divide by -3
- 4. Multiple-choice 30 seconds 1 pt Solve for x: 8x - 10 = -34 4 3 -3 -24
- 6. Multiple-choice 1 minute 1 pt Solve for the variable. 3x=-27 x=9 x=-30 x=-9 x=-24
- 7. Multiple-choice 2 minutes 1 pt Solve for "x" ? x-9=15 23 6 24 11
- 8. Multiple-choice 3 minutes 1 pt Solve for " x" ? 4x-6=30 9 13 36 4
- 9. Multiple-choice 30 seconds 1 pt h/3=7 10 4 21 2.33
- 10. Multiple-choice 3 minutes 1 pt 2x - 3 + 4x = 27 12 4 5 15
- 11. Multiple-choice 3 minutes 1 pt y - 14 = 66 80 52 79 Answer Not Given
- 12. Multiple-choice 3 minutes 1 pt Solve the equation: 4p = 16 p = 12 p = 64 p = 3 p = 4
- 13. Multiple-choice 30 seconds 1 pt Ted bought a dolphin poster for $12. He now has $5. Write and solve a subtraction equation to find out how much money Ted took to the aquarium. m - 12 = 5 m - 5 = 12 m + 5 = 12 m + 12 = 5
- 14. Multiple-choice 30 seconds 1 pt Cheryl gets paid $8 per hour at her job at the record store. She made a total of $96 last week. Write a multiplication equation to show how many hours she worked last week. 96h = 8 h/96 = 8 8h = 96 h/8 = 96
- 15. Multiple-choice 30 seconds 1 pt Translate: Twelve less than a number 12 - x x - 12 12x x + 12
- 16. Multiple-choice 15 minutes 1 pt Solve: 2x + 1 = 5 3 2 -2 -3
- 17. Multiple-choice 1 minute 1 pt Solve the equation x+8=-22 30 14 -14 -30

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## Completing the Square

" Completing the Square " is where we ...

But if you have time, let me show you how to " Complete the Square " yourself.

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x 2 + bx can be rearranged nearly into a square ...

... and we can complete the square with (b/2) 2

In Algebra it looks like this:

So, by adding (b/2) 2 we can complete the square.

The result of (x+b/2) 2 has x only once , which is easier to use.

## Keeping the Balance

So let's see how to do it properly with an example:

And now x only appears once, and our job is done!

## A Shortcut Approach

Here is a quick way to get an answer. You may like this method.

First think about the result we want: (x+d) 2 + e

After expanding (x+d) 2 we get: x 2 + 2dx + d 2 + e

Now see if we can turn our example into that form to discover d and e

## Example: try to fit x 2 + 6x + 7 into x 2 + 2dx + d 2 + e

- We know that 6x must end up as 2dx , so d must be 3
- Next we see that 7 must become d 2 + e = 9 + e , so e must be −2

And we get the same result (x+3) 2 − 2 as above!

Now, let us look at a useful application: solving Quadratic Equations ...

## Solving General Quadratic Equations by Completing the Square

We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have a coefficient of a in front of x 2 :

But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:

Now we can solve a Quadratic Equation in 5 steps:

- Step 1 Divide all terms by a (the coefficient of x 2 ).
- Step 2 Move the number term ( c/a ) to the right side of the equation.
- Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p) 2 = q, which can be solved rather easily:

- Step 4 Take the square root on both sides of the equation.
- Step 5 Subtract the number that remains on the left side of the equation to find x .

## Example 1: Solve x 2 + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x 2 is 1

Step 2 Move the number term to the right side of the equation:

Step 4 Take the square root on both sides of the equation:

Step 5 Subtract 2 from both sides:

## Example 2: Solve 5x 2 – 4x – 2 = 0

(b/2) 2 = (0.8/2) 2 = 0.4 2 = 0.16

Step 5 Subtract (-0.4) from both sides (in other words, add 0.4):

## Why "Complete the Square"?

Why complete the square when we can just use the Quadratic Formula to solve a Quadratic Equation?

Also Completing the Square is the first step in the Derivation of the Quadratic Formula

Just think of it as another tool in your mathematics toolbox.

## Footnote: Values of "d" and "e"

How did I get the values of d and e from the top of the page?

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