Completing the Square — Method, Formula, Examples

TL;DR

Completing the square rewrites a quadratic $$ax^2 + bx + c$$ in the form $$a(x+h)^2+k$$ by adding and subtracting $$\left(\frac{b}{2a}\right)^2$$. This article covers the four-step method, three worked examples at Quick/Standard/Stretch tiers, the geometric meaning, and how completing the square produces the quadratic formula itself.

A Method That Turns Any Quadratic Into a Perfect Square

Every quadratic expression hides a perfect square inside it. Finding that square — by adjusting one term — is what completing the square does.

The Formula

For a quadratic $$ax^2 + bx + c$$: $$ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^{2} + c - \frac{b^{2}}{4a}.$$

The term $$\left(\frac{b}{2a}\right)^{2}$$ is the missing corner — the small square that turns the L-shaped piece $$x^2 + \frac{b}{a}x$$ into a complete $$\left(x + \frac{b}{2a}\right)^{2}$$ square.

Quick facts.

• Identity: $$x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$$ (when $$a=1$$).
• Use cases: solving any quadratic, finding the vertex of a parabola, deriving the quadratic formula.
• Geometric origin: literally completing a partial square into a whole square.
• Grade introduced: CBSE Class 10; CCSS-M HSA-REI.B.4.a.

The Four-Step Method of Completing the Square

1. Move the constant. Get the quadratic to the form $$ax^2 + bx = -c$$.
2. Divide by $$a$$ if needed. If the leading coefficient is not 1, divide every term by $$a$$.
3. Add $$\left(\frac{b}{2}\right)^2$$ to both sides. Half the coefficient of $$x$$, square it, add to both sides. The left side is now a perfect square trinomial.
4. Write the left side as $$\left(x + \frac{b}{2}\right)^{2}$$ and solve. Take the square root and solve for $$x$$.

Three Worked Examples of Completing the Square

Quick. Solve $$x^2 + 6x = 7$$.

Half of 6 is 3; square is 9. Add 9 to both sides.

$$x^2 + 6x + 9 = 16.$$
$$(x + 3)^2 = 16.$$
$$x + 3 = \pm 4.$$
Final answer: $$x = 1$$ or $$x = -7$$.

Standard (Wrong-Path First — The Tempting Shortcut That Doesn't Work). Solve $$x^2 - 10x + 16 = 0$$.

The memorizer recalls "add $$\left(\frac{b}{2}\right)^2$$" and writes $$x^2 - 10x + 25 = 25$$.

$$x^2 - 10x + 25 = 25.$$
Check: $$0^2 - 10(0) + 16 = 16 \neq 0$$. Wrong.

Final answer: $$x = 8$$ or $$x = 2$$.

Stretch. Solve $$2x^2 - 12x + 7 = 0$$ by completing the square.

Divide every term by 2 first:

$$x^2 - 6x + \frac{7}{2} = 0.$$ Move constant.
$$x^2 - 6x = -\frac{7}{2}.$$
Add $$9$$:
$$x^2 - 6x + 9 = -\frac{7}{2} + 9 = \frac{11}{2}.$$ Final answer: $$x = 3 \pm \frac{\sqrt{22}}{2}$$.

Why Completing the Square Matters — Beyond Solving

Completing the square does three jobs at once:

• It solves any quadratic.
• It produces the vertex form.
• It derives the quadratic formula.

Completing the Square: Mistakes Worth Walking Through

1. Forgetting to add to both sides.

2. Forgetting to divide by $$a$$ when $$a \neq 1$$.

3. Mishandling the sign on $$\frac{b}{2}$$.

4. Confusing the perfect square with the original quadratic.

The Mathematicians Who Shaped Completing the Square

• Babylonian scribes (c. 1800 BCE, Mesopotamia).
• Al-Khwarizmi (c. 780–850 CE, Persia).
• René Descartes (1596–1650, France).

Conclusion

• Completing the square rewrites a quadratic as a perfect-square trinomial plus a constant by adding $$\left(\frac{b}{2}\right)^2$$ to both sides.
• The technique produces both the roots and the vertex of a parabola.

Try It Yourself — Three Problems

1. Solve $$x^2 + 8x = 9$$ by completing the square.
2. Solve $$2x^2 - 8x + 3 = 0$$ by completing the square.
3. Rewrite $$y = x^2 - 6x + 11$$ in vertex form and find the vertex.