Completing the Square: Another Way to Solve Quadratic Equation

 

Completing the Square: Another Way to Solve Quadratic Equations

A friendly, visual guide for students, teachers, homeschool families, and anyone building algebra confidence.

Big idea: Completing the square turns part of a quadratic into a perfect square, making the equation easier to solve by taking square roots.

What Does “Complete the Square” Mean?

A perfect-square trinomial has the pattern x² + 2px + p² = (x + p)². For example, x² + 6x + 9 = (x + 3)².

To create the missing constant, take half of the coefficient of x, then square that result. Whatever you add to one side of an equation, add to the other side too.

The Key Identity

x² + 2px + p² = (x + p)²

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The Four-Step Process

1. Make the coefficient of x² equal to 1. Divide the equation by the leading coefficient when needed.
2. Move the constant. Leave the x terms on one side.
3. Add the square of half the x coefficient to both sides.
4. Write a perfect square, take square roots, and solve. Remember both the positive and negative roots.

Worked Example 1: Integer Solutions

Solve: x² + 6x + 5 = 0

Move the constant: x² + 6x = −5.

Half of 6 is 3, and 3² = 9. Add 9 to both sides:

x² + 6x + 9 = −5 + 9
(x + 3)² = 4

Take both square roots: x + 3 = 2 or x + 3 = −2.

Solutions: x = −1 and x = −5

Worked Example 2: A Fraction Appears

Solve: 2x² + 8x − 3 = 0

Divide by 2: x² + 4x − 3 ÷ 2 = 0, so x² + 4x = 3 ÷ 2.

Half of 4 is 2, and 2² = 4. Add 4 to both sides:

x² + 4x + 4 = 3 ÷ 2 + 4 = 11 ÷ 2
(x + 2)² = 11 ÷ 2

Take both square roots and subtract 2:

Solutions: x = −2 + √(11 ÷ 2) and x = −2 − √(11 ÷ 2)

Worked Example 3: A Leading Coefficient

Solve: 3x² + 12x + 6 = 0

Divide by 3: x² + 4x + 2 = 0, so x² + 4x = −2.

Add 4 to both sides: x² + 4x + 4 = 2, which becomes (x + 2)² = 2.

Take square roots:

Solutions: x = −2 + √2 and x = −2 − √2

A Memory Tip

Half it, square it, add it, root it.
Take half the x coefficient, square it, add it to both sides, then take both roots.

Common Mistakes to Avoid

• Adding to only one side: Keep the equation balanced by adding the same value to both sides.
• Using the full coefficient instead of half: For 8x, use half of 8, which is 4, then square 4.
• Forgetting the second root: √k means the solutions can be +√k and −√k.
• Skipping the leading coefficient step: The coefficient of x² should be 1 before forming the square.
• Changing the sign incorrectly: Move constants carefully and check by substitution.

Practice Questions

Complete the square and solve each equation. Give exact answers when a square root remains.

Question	Solve
1	x2 − 8x + 7 = 0
2	x2 + 10x + 9 = 0
3	3x2 + 12x + 6 = 0
4	2x2 − 4x − 6 = 0

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Answers and Quick Checks

1. (x − 4)² = 9, so x = 1 or x = 7.

2. (x + 5)² = 16, so x = −1 or x = −9.

3. Divide by 3, then (x + 2)² = 2, so x = −2 + √2 or x = −2 − √2.

4. Divide by 2, then (x − 1)² = 4, so x = −1 or x = 3.

Final Takeaway

Completing the square transforms a quadratic into a perfect-square equation that you can solve with square roots.

Half it → Square it → Add it → Root it