Completing the Square Formula

The Formula

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

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

Quick Example

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

Notation

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

What This Formula Means

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

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

Formal View

x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + \left(c - \frac{b^2}{4}\right). More generally, ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}, valid \forall a \neq 0.

Worked Examples

Example 1

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

Solution

  1. 1
    Take half of the coefficient of x: \frac{6}{2} = 3.
  2. 2
    Square it: 3^2 = 9.
  3. 3
    Add and subtract 9: x^2 + 6x + 9 - 9 + 2 = (x + 3)^2 - 7.

Answer

(x + 3)^2 - 7
Completing the square transforms a quadratic into vertex form a(x-h)^2 + k by creating a perfect square trinomial. The vertex is (-3, -7).

Example 2

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

Common Mistakes

  • Forgetting to subtract the same value you added (you must add AND subtract \left(\frac{b}{2}\right)^2 to maintain equality)
  • Not factoring out a first when a \neq 1
  • Arithmetic errors when computing \left(\frac{b}{2}\right)^2, especially with fractions or negatives

Why This Formula Matters

Completing the square converts standard form to vertex form, derives the quadratic formula, and appears throughout higher math including conic sections and integral techniques.

Frequently Asked Questions

What is the Completing the Square formula?

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

How do you use the Completing the Square formula?

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

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

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

Why is the Completing the Square formula important in Math?

Completing the square converts standard form to vertex form, derives the quadratic formula, and appears throughout higher math including conic sections and integral techniques.

What do students get wrong about Completing the Square?

When a \neq 1, you must factor out a from the x^2 and x terms first, then complete the square inside the parentheses—and be careful when distributing a back.

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 →
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