Completing the square explained
Assumption: this covers equations of the form x^2 + bx + c = 0 (where the number in front of x^2 is 1), which matches her homework problem. Her course may write steps slightly differently, so follow the teacher's version.
The short version
Completing the square means rewriting x^2 + bx as a perfect square by adding (b/2)^2. Once one side is "something squared," you can solve by taking a square root.
The big idea
Your daughter already knows the quadratic formula. Completing the square is actually where the quadratic formula comes from, so this method explains the formula she already uses.
The problem: In an equation like x^2 + 6x − 7 = 0, the x shows up twice (as x^2 and as 6x). You can't easily get x by itself when it's in two places.
The goal: Squish both x-terms into one package, like (x + something)^2. Then x appears only once, and you can undo the square with a square root.
Why (b/2)^2? Here's the picture in words. Picture x^2 as a square tile, x long on each side. Picture bx as a long rectangle, x tall and b wide.
- Cut the bx rectangle in half lengthwise. You get two strips, each x by b/2.
- Attach one strip to the right side of the square and one strip to the bottom.
- Now you have almost a bigger square, with sides x + b/2. But there's a missing corner in the bottom-right.
- That missing corner is a small square, b/2 on each side. Its area is (b/2)^2.
Adding (b/2)^2 literally fills in the missing corner. That's why it's called completing the square.
Why add it to both sides? An equation is a balance scale. If you add a corner piece to the left side only, the scale tips and the equation is no longer true. Adding the same amount to the right side keeps everything balanced.
Think of it like...
A puzzle frame with one missing corner piece. You've laid out the big square and the two edge strips, and there's an obvious square-shaped hole in the corner. You know exactly what size piece fits: (b/2) by (b/2). Snap it in, and the picture is complete.
Where it breaks down: with real puzzles you can't add pieces from nowhere. In algebra you can, but only if you add the same amount to the other side to keep things fair. Also, when b is negative, the "strips" get subtracted rather than attached, so the picture is harder to draw, but the arithmetic works the same way.
Key terms
| Term | What it means | Quick example |
|---|---|---|
| Quadratic equation | An equation whose highest power of x is 2 | x^2 + 8x − 20 = 0 |
| Coefficient | The number multiplied by a variable | In 8x, the coefficient is 8 |
| Constant term | The plain number with no x attached | In x^2 + 8x − 20, it's −20 |
| Perfect square trinomial | A three-term expression that equals something squared | x^2 + 8x + 16 = (x + 4)^2 |
| Binomial squared | A two-term expression times itself | (x + 4)^2 = (x + 4)(x + 4) |
| ± (plus or minus) | Shows that a square root has two answers, positive and negative | If y^2 = 36, then y = ±6 |
Worked example
A quick note: to keep things fair for her homework, this is a similar problem with different numbers. The hints afterward will help her work through her own problem.
Solve x^2 + 8x − 20 = 0 by completing the square.
Move the constant to the right side. Add 20 to both sides: x^2 + 8x = 20. Why: we want only the x-terms on the left so there's room to build our square.
Find half of b. Here b = 8, so b/2 = 4. Why: these are the two strips, each 4 wide.
Square it. (b/2)^2 = 4^2 = 16. Why: this is the area of the missing corner.
Add it to both sides. x^2 + 8x + 16 = 20 + 16, so x^2 + 8x + 16 = 36. Why: fill the corner on the left and keep the scale balanced.
Rewrite the left side as a square. (x + 4)^2 = 36. Why: the number inside the parentheses is always b/2 (here, 4). Check it: (x + 4)(x + 4) = x^2 + 8x + 16. ✓
Take the square root of both sides, using ±. x + 4 = ±6. Why: both 6^2 and (−6)^2 equal 36.
Solve both cases.
- x + 4 = 6, so x = 2
- x + 4 = −6, so x = −10
Check.
- x = 2: 4 + 16 − 20 = 0 ✓
- x = −10: 100 − 80 − 20 = 0 ✓
Hints for her own problem (x^2 + 6x − 7 = 0)
- What does she need to do to both sides to move the −7 off the left side?
- Her b is 6. What is half of 6, and what is that number squared? That's her "corner piece." Can she draw the x-by-x square with two strips and point to the gap?
- After adding the corner piece to both sides, what goes inside the parentheses in (x + ___)^2? And did she remember the ± when taking the square root?
She can check her own answers by plugging them back into the original equation. She can also check them with the quadratic formula, since she already knows it.
Common mistakes
- Adding the corner to only one side. Instead: say "both sides" out loud every time you add (b/2)^2.
- Adding b/2 instead of (b/2)^2. Instead: remember the corner is a square, so its area is side times side. Halve first, then square.
- Forgetting the ± when taking the square root. Instead: write ± immediately, and expect two answers.
- Not moving the constant first. Instead: get the equation to the form x^2 + bx = (number) before doing anything else.
- Losing the sign when b is negative. Instead: for x^2 − 10x, half of b is −5, so the square is (x − 5)^2. The squared corner is still +25.
Check yourself
- What number completes the square for x^2 + 10x? Write the result as a binomial squared.
- What number completes the square for x^2 − 4x? Write the result as a binomial squared.
- Solve x^2 + 4x − 12 = 0 by completing the square.
- Solve x^2 − 2x − 4 = 0 by completing the square. (Hint: the answer won't be a whole number.)
- What number completes the square for x^2 + 3x? Write the result as a binomial squared.
- In your own words: Using the tile picture, explain why we add (b/2)^2, and why we add it to