Completing the Square

Master the technique of completing the square to solve quadratic equations and rewrite them in vertex form.

Advanced30 minLesson
Printable worksheet

Definition

Completing the square is a technique to rewrite a quadratic expression ax2+bx+cax^2 + bx + c into the form a(x−h)2+ka(x - h)^2 + k, called vertex form.
The key insight: we add and subtract the same value to create a perfect square trinomial.
x2+bx+(b2)2=(x+b2)2x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2
For example:
x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2
Here, (62)2=9\left(\frac{6}{2}\right)^2 = 9 completes the square.

Try it now

What value completes the square for x2+6xx^2 + 6x?

Worked Examples

Complete the square: x2+8xx^2 + 8x

1

Identify the coefficient of x

b=8b = 8 → Coefficient is 8

2

Take half of b

82=4\frac{8}{2} = 4 → Half is 4

3

Square the result

42=164^2 = 16 → Need to add 16

4

Add and subtract this value

x2+8x+16−16x^2 + 8x + 16 - 16 → Balance maintained

5

Factor the perfect square trinomial

(x+4)2−16(x + 4)^2 - 16 → Completed square form

Common Mistakes

Forgetting to add to both sides when solving an equation

Why it's wrong: When completing the square in an equation, whatever you add to one side must be added to the other side to maintain equality.

Correct: After adding (b2)2\left(\frac{b}{2}\right)^2 to the left, add the same value to the right.

Not factoring out the leading coefficient first

Why it's wrong: When a≠1a \neq 1 in ax2+bx+cax^2 + bx + c, completing the square directly gives wrong results.

Correct: First factor out aa from the x2x^2 and xx terms, then complete the square inside.

Using bb instead of b2\frac{b}{2} to complete the square

Why it's wrong: The formula requires half of the coefficient of xx, squared.

Correct: Always use (b2)2\left(\frac{b}{2}\right)^2, not b2b^2.

Forgetting to multiply when distributing

Why it's wrong: When aa is factored out, the value subtracted inside gets multiplied by aa.

Correct: In 2(x2+6x+9−9)2(x^2 + 6x + 9 - 9), the −9-9 becomes −18-18 when distributed: 2(−9)=−182(-9) = -18.

Interactive Visual

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise

Area model: completing the square

x² + bx is a square with a missing corner. Adding (b/2)² completes it.

x2x^2
3x3x
3x3x
+9+9
xx
33
xx
33

Length × width = area

x2+6x=(x+3)2−9x^2 + 6x = (x + 3)^2 - 9

Add 9 to fill the corner and get a perfect square.

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Practice Problems

16 problems
Problem 1 of 16
Easy

What value completes the square for x2+6xx^2 + 6x?

Why It Matters

Completing the square is one of the most powerful algebraic techniques because it:
  • Solves any quadratic equation (even when factoring doesn't work)
  • Reveals the vertex of a parabola directly from the equation
  • Derives the quadratic formula (which comes from completing the square!)
  • Transforms circles from general to standard form in geometry
Without this technique, many quadratic equations would be unsolvable by hand!

Real World Applications

Projectile Motion

Engineers use completing the square to find the maximum height of projectiles.

Example:

The height of a ball is h=−16t2+64t+5h = -16t^2 + 64t + 5 feet. Completing the square: h=−16(t−2)2+69h = -16(t - 2)^2 + 69. Maximum height is 69 feet at t=2t = 2 seconds.

1Try It Yourself

A rocket's height is given by h=−5t2+30t+10h = -5t^2 + 30t + 10 meters.

What is the maximum height?

Step 1: Write the mathematical expression

Complete the square to find the vertex:

Architecture and Design

Architects use completing the square to design parabolic arches and determine their highest points.

Example:

A bridge arch follows y=−0.01x2+2xy = -0.01x^2 + 2x. Converting to vertex form: y=−0.01(x−100)2+100y = -0.01(x - 100)^2 + 100. The arch reaches 100 meters high at its center.

2Try It Yourself

A parabolic antenna dish is modeled by y=−0.25x2+4x+3y = -0.25x^2 + 4x + 3.

At what x-coordinate is the dish deepest?

Step 1: Write the mathematical expression

Find the vertex x-coordinate:

Economics and Profit Optimization

Businesses use quadratic models to find optimal pricing for maximum profit.

Example:

A company's profit is P=−2x2+40x−150P = -2x^2 + 40x - 150 dollars where xx is the price. Vertex form: P=−2(x−10)2+50P = -2(x - 10)^2 + 50. Maximum profit of 50 dollars occurs at price 10 dollars.

3Try It Yourself

Revenue is modeled by R=−3x2+24xR = -3x^2 + 24x where xx is units sold in thousands.

How many units maximize revenue?

Step 1: Write the mathematical expression

Find the vertex:

Key Takeaways

  • 1Completing the square transforms x2+bxx^2 + bx into (x+b2)2−(b2)2(x + \frac{b}{2})^2 - (\frac{b}{2})^2
  • 2The key value to add is always (b2)2\left(\frac{b}{2}\right)^2
  • 3When a≠1a \neq 1, factor out aa from the xx-terms first
  • 4Vertex form a(x−h)2+ka(x - h)^2 + k reveals the vertex (h,k)(h, k) directly
  • 5This technique can solve any quadratic equation, even when factoring fails

Frequently Asked Questions

Use completing the square when: (1) the quadratic doesn't factor nicely with integers, (2) you need to find the vertex of a parabola, or (3) you're deriving the quadratic formula. Factoring is faster when it works, but completing the square always works.
Use completing the square when: (1) the quadratic doesn't factor nicely with integers, (2) you need to find the vertex of a parabola, or (3) you're deriving the quadratic formula. Factoring is faster when it works, but completing the square always works.
Adding and subtracting the same value is like adding zero, which doesn't change the expression's value. This lets us create a perfect square trinomial without changing what the expression equals.
The quadratic formula is derived by completing the square on the general form ax2+bx+c=0ax^2 + bx + c = 0. The formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} comes directly from this process!

Glossary

Completing the square
A method to rewrite a quadratic expression as a perfect square plus or minus a constant
Perfect square trinomial
A trinomial that factors as (x+a)2(x + a)^2 or (x−a)2(x - a)^2, such as x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2
Vertex form
The form a(x−h)2+ka(x - h)^2 + k where (h,k)(h, k) is the vertex of the parabola
Standard form
The form ax2+bx+cax^2 + bx + c for a quadratic expression

Formula Card

Completing the Square Formula

x2+bx=(x+b2)2−(b2)2x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2

The core transformation for completing the square

Value to Complete

(b2)2\left(\frac{b}{2}\right)^2

The value you add and subtract to create a perfect square trinomial

Vertex Form

y=a(x−h)2+ky = a(x - h)^2 + k

The vertex of the parabola is at point (h, k)

Perfect Square Patterns

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

Expanding a squared binomial creates a perfect square trinomial

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