Completing the Square Formula

Completing the square is a technique for rewriting ax^2 + bx + c in vertex form a(x.

The Formula

x2+bx+c=(x+b2)2βˆ’b24+cx^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c Add and subtract (b2)2\left(\frac{b}{2}\right)^2 to complete the square.

When to use: Imagine you have x2+6xx^2 + 6x and want a perfect square. A perfect square like (x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9 needs that extra +9+9. So you add 9 and subtract 9 to keep the expression equalβ€”then group the perfect square part.

Quick Example

x2+6x+2=(x2+6x+9)βˆ’9+2=(x+3)2βˆ’7x^2 + 6x + 2 = (x^2 + 6x + 9) - 9 + 2 = (x + 3)^2 - 7
Vertex form with vertex (βˆ’3,βˆ’7)(-3, -7).

Notation

(b2)2\left(\frac{b}{2}\right)^2 is the value added and subtracted. The result is a perfect square (x+b2)2(x + \frac{b}{2})^2 plus a constant.

What This Formula Means

A technique for rewriting ax2+bx+cax^2 + bx + c in vertex form a(xβˆ’h)2+ka(x - h)^2 + k by adding and subtracting the value (b2a)2\left(\frac{b}{2a}\right)^2 to create a perfect square trinomial.

Imagine you have x2+6xx^2 + 6x and want a perfect square. A perfect square like (x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9 needs that extra +9+9. So you add 9 and subtract 9 to keep the expression equalβ€”then group the perfect square part.

Formal View

x2+bx+c=(x+b2)2+(cβˆ’b24)x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + \left(c - \frac{b^2}{4}\right). More generally, ax2+bx+c=a(x+b2a)2+cβˆ’b24aax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}, valid βˆ€aβ‰ 0\forall a \neq 0.

Worked Examples

Example 1

medium
Rewrite x2+6x+2x^2 + 6x + 2 in vertex form by completing the square.

Answer

(x+3)2βˆ’7(x + 3)^2 - 7

First step

1
Take half of the coefficient of xx: 62=3\frac{6}{2} = 3.

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Example 2

hard
Solve x2βˆ’4xβˆ’5=0x^2 - 4x - 5 = 0 by completing the square.

Example 3

medium
Rewrite x2βˆ’10x+18x^2 - 10x + 18 in vertex form by completing the square.

Common Mistakes

  • Adding (b2)2(\tfrac{b}{2})^2 without also subtracting it - you must keep the expression equal; add and subtract the same amount.
  • Adding (b/2)2(b/2)^2 without first factoring out a leading coefficient aβ‰ 1a\ne 1 β€” factor out aa, complete the square inside, then distribute.
  • Halving bb but not squaring it - the value added is the SQUARE of half the middle coefficient.

Why This Formula Matters

It is the universal method behind vertex form and the quadratic formula itself, and it works on every quadratic, even those that refuse to factor. Skipping the 'add AND subtract' balance is what makes students change the expression's value by accident. Recognizing it by "Am I manufacturing a perfect-square trinomial by adding and subtracting the same value to keep it equal?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from factoring and quadratic formula and reading vertex form in a mixed problem set.

Frequently Asked Questions

What is the Completing the Square formula?

A technique for rewriting ax2+bx+cax^2 + bx + c in vertex form a(xβˆ’h)2+ka(x - h)^2 + k by adding and subtracting the value (b2a)2\left(\frac{b}{2a}\right)^2 to create a perfect square trinomial.

How do you use the Completing the Square formula?

Imagine you have x2+6xx^2 + 6x and want a perfect square. A perfect square like (x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9 needs that extra +9+9. So you add 9 and subtract 9 to keep the expression equalβ€”then group the perfect square part.

What do the symbols mean in the Completing the Square formula?

(b2)2\left(\frac{b}{2}\right)^2 is the value added and subtracted. The result is a perfect square (x+b2)2(x + \frac{b}{2})^2 plus a constant.

Why is the Completing the Square formula important in Math?

It is the universal method behind vertex form and the quadratic formula itself, and it works on every quadratic, even those that refuse to factor. Skipping the 'add AND subtract' balance is what makes students change the expression's value by accident. Recognizing it by "Am I manufacturing a perfect-square trinomial by adding and subtracting the same value to keep it equal?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from factoring and quadratic formula and reading vertex form in a mixed problem set.

What do students get wrong about Completing the Square?

The procedure for completing the square is the easy part; the trap is adding (b2)2(\tfrac{b}{2})^2 without also subtracting it. Asking "Am I manufacturing a perfect-square trinomial by adding and subtracting the same value to keep it equal?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Completing the Square formula?

Before studying the Completing the Square formula, you should understand: quadratic standard form, expressions.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula β†’
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