Reshaping any quadratic into a perfect square so you can take a root.
Back in 11.1 you could solve a quadratic only when it already looked like a perfect square sitting next to a number — something tidy like (x − 3)2 = 16. You took square roots of both sides and walked away with the answer. The trouble is that almost no quadratic shows up dressed that way. x2 + 6x + 5 = 0 has no obvious square in it at all. So we need a move that takes any quadratic and reshapes it into that friendly perfect-square form. That move is completing the square, and it is exactly what it sounds like: you have a shape that is almost a square, and you add the one missing piece to finish it.
By the end of this lesson you'll be able to look at x2 + bx and instantly name the number that completes it, rewrite any quadratic as (x + m)2 = k, and solve the whole equation from there — even when the leading coefficient isn't 1. Throughout, the colors carry the meaning: blue is the square, the x2 block, teal is the linear term bx, the strips along the sides, amber is the constant — and the little corner number we add, and violet marks the roots, the answers we're hunting for.
You already know how to multiply out a perfect square. From factoring:
(x + m)2 = x2 + 2mx + m2
Read that left to right and it's old news. The whole trick of this lesson is to read it right to left — backward. Suppose someone hands you just the front two pieces, x2 + 2mx, and asks: what's missing to make this a perfect square? Stare at the middle term. The coefficient of x is 2m. So m is just half the coefficient of x. And the missing last piece is m2 — which is that same half, squared.
If you see x2 + (coeff)·x, the piece that finishes the perfect square is (coeff⁄2)2. Take half the x-coefficient, square it. That's the whole formula.
Let's make it concrete. Take x2 + 6x. Half of 6 is 3, and 3 squared is 9. So the number that completes the square is 9, and indeed
x2 + 6x + 9 = (x + 3)2
Notice the 3 inside the parentheses is the same 3 you got from halving — that's not a coincidence, it's the whole reason the recipe works. Half-of-the-x-coefficient lands inside the square; that same number squared is what you add on the outside.
Drag m and watch the area split into the four pieces of (x + m)2. The amber corner is the piece that completes the square.
Now we turn that observation into a habit you can run on autopilot. Given any x2 + bx, here is the recipe in three words: half it, square it, add it.
Three quick reps. For x2 + 8x: half of 8 is 4, squared is 16, so we get x2 + 8x + 16 = (x + 4)2. The sign of b rides straight through: for x2 − 8x, half of −8 is −4, and (−4) squared is still 16 (squaring kills the minus), giving x2 − 8x + 16 = (x − 4)2. The number you add is always positive; the sign only shows up inside the parentheses.
Odd coefficients don't break anything — you just get a fraction. For x2 + 5x: half of 5 is 5/2, squared is 25/4, so x2 + 5x + 25/4 = (x + 5/2)2. Don't let the fraction scare you; the recipe is identical.
The number you add is (b/2)2, not b2. For x2 + 6x, you add 9, not 36. If you ever find yourself adding the full coefficient squared, you've forgotten to halve first. The picture in the next section shows exactly why halving is forced on you — there are two strips to fold in, not one.
One honest word about language. When we're just rewriting an expression, "add it" means literally writing the corner in to build the square. When we're solving an equation, you can't add a number to one side for free — you have to add it to both sides to keep the equation balanced. We'll lean on that in 11.2.4. For now, just get fluent at naming the corner.
Step b up and down. The dashed amber square is the corner you must add to finish x2 + bx.
Everything so far assumed the squared term was a bare x2 with a coefficient of 1. The recipe needs that. So when the leading coefficient — call it a — is anything else, your first move is to get rid of it. There are two clean ways, and which you pick depends on whether you're rewriting an expression or solving an equation.
Solving an equation? Divide every term by a. Since the whole thing equals zero, dividing both sides by a changes nothing and instantly gives you a leading 1. Watch 2x2 + 12x + 10 = 0:
Just rewriting an expression? Factor a out of the terms that contain x, then complete the square inside the parentheses. For 3x2 − 12x:
For solving equations — which is most of what we'll do — the divide-by-a path is almost always less error-prone, because you never have to track an extra factor multiplying a parenthesis. When in doubt, divide first.
Now we put it all together and actually solve. The plan has a rhythm worth memorizing: get the constant out of the way, add the corner to both sides, fold the left into a square, take roots. Let's solve x2 + 6x + 5 = 0 from start to finish.
Two things to underline. First, the step that beginners skip: adding the corner to both sides. If you add 9 only on the left, you've changed the equation and your answer will be wrong. Second, the ± when you take the root — a positive number has two square roots, and dropping the negative one throws away a perfectly good solution. Most "I only got one answer" mistakes trace back to a missing ±.
After folding, you'll have (x + m)2 = k. A square can never be negative, so if k < 0 there is no real solution — no real number squares to a negative. For example (x + 2)2 = −9 simply has no real answer. That's not a mistake on your part; it's the equation telling you its parabola never touches the x-axis.
Set b and c for x2 + bx + c = 0. Watch it fold into (x + b/2)2 = k and read off the roots.
The name "completing the square" is not a metaphor. It is a literal description of a picture, and once you see it you'll never forget why we halve and square. Start with x2 + bx. Read it as area: a square of side x (area x2) sitting next to a rectangle that is b tall and x wide (area bx).
Here's the move. Slice that b-by-x rectangle down the middle into two strips, each one b/2 wide and x long. Lay one strip along the right edge of the square and the other along the bottom edge. Look at what you've built: it's almost a bigger square with side (x + b/2) — but there's a small square notch missing in the far corner, exactly b/2 by b/2.
Toggle the corner on and off. "Before" is x2 + bx; "after" is the completed square (x + b/2)2.
Now the punchline. To turn that notched shape into an honest, complete square you must drop in the missing corner — a tile of area (b/2)2. Add that corner and you have "completed the square." That's the entire name. And the picture explains the formula you might otherwise just memorize: the strips are each b/2 wide because there are two of them sharing the rectangle, so the corner they leave open is b/2 on a side, with area (b/2)2. That's why we halve before we square — and why adding the full b2 would be wildly too big, a corner four times larger than the gap.
Reading the perfect-square formula backward tells you that x2 + bx is missing exactly one piece, the corner (b/2)2 — half the x-coefficient, squared. Add it and the expression collapses into (x + b/2)2. If the leading coefficient isn't 1, divide it out first (or factor it out when you're only rewriting). To solve an equation you move the constant aside, add the corner to both sides, fold the left into a square, and take roots with a ± — landing on the two roots, unless the right side came out negative, in which case there are none. And the whole thing is just geometry: two strips around a square leave one corner open, and you fill it.
We're about to do the boldest thing in the chapter: run completing the square not on numbers but on the letters ax2 + bx + c = 0 themselves. The corner becomes (b/2a)2, the algebra folds, and out drops the quadratic formula −b ± √(b2−4ac)2a — not as a thing to memorize, but as a thing you built.
Work each one out first, then open the answer to check your thinking.
Tap the answer you think is right.
This lesson targets CCSS A-REI.B.4a (solve quadratics by completing the square) and A-SSE.B.3b (complete the square to reveal the vertex / maximum or minimum). Two misconceptions dominate and both are addressed head-on: students add b2 instead of (b/2)2, and they forget to add the corner to both sides when solving an equation. The gnomon / area model is the antidote to the first — it makes "halve, then square" visually inevitable, since the rectangle must split into two strips. The "take roots with ±, watch for k < 0" framing seeds later work with the discriminant and with vertex form for graphing parabolas. If a student can narrate the geometric picture in their own words, they own the method rather than memorizing it.