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Updated: 2026-03-18

Master Completing the Square: Practice Problems & Solved Examples

Overview: This article serves as a key resource for high school algebra. It focuses on mastering the method of solving quadratic equations by completing the square, one of the most important techniques in the curriculum. The guide breaks down the process of transforming a general quadratic equation into a perfect square trinomial, providing clear, step-by-step examples and practice problems.

Master the Completing the Square Method

Welcome to your comprehensive resource for mastering the technique of solving quadratic equations by completing the square. This fundamental algebra skill is crucial for high school mathematics success. Explore our detailed examples and targeted practice problems below to build your expertise and confidence in this method.

Understanding How to Solve Quadratic Equations by Completing the Square

In algebra courses, students must learn to solve general quadratic equations using the "completing the square" approach. A standard quadratic equation involves a second-degree polynomial and is expressed as:

ax² + bx + c = d

Here, a, b, c, and d represent real number coefficients. This equation can always be simplified. The goal is to have a leading coefficient of 1 and to move the constant term to the equation's right side, setting it to zero. Achieve this by subtracting d from both sides and then dividing the entire equation by a:

x² + (b/a)x + (c-d)/a = 0

Therefore, learning to solve a simplified equation of this form is essential:

x² + bx + c = 0

To complete the square, add the term (b²/4) to both sides of the equation:

x² + bx + b²/4 = -c + b²/4

The left side is now a perfect square trinomial:

(x + b/2)² = -c + b²/4

Finally, you can take the square root of both sides. This yields |x + b/2| and allows you to solve for x through basic algebra. If the right side is a negative number, the equation possesses no real-number solutions, though complex number solutions do exist.

Test Your Skills: A Completing the Square Quiz

Determine the necessary term to add to each expression below to create a perfect square trinomial.

x² + 2x
x² - 6x
x² + 3x
x² + 6x + 6
x² - 2x + 3

Quiz Answers and Explanations

Add 1 to form the trinomial (x+1)².
Add 9 to form the trinomial (x-3)².
Add 9/4 to form the trinomial (x + 3/2)².
Add 3 to form the trinomial (x+3)².
Subtract 2 to form the trinomial (x-1)².

Step-by-Step Solved Examples

Learn the completing the square method through the following detailed examples.

Example 1: Solve x² - x + 0.25 = 1

First, move the constant: x² - x + 0.25 - 1 = 0 which simplifies to x² - x - 0.75 = 0. To complete the square for x² - x, take half of -1, which is -0.5, and square it to get 0.25. Add and subtract this value:

(x² - x + 0.25) - 0.25 - 0.75 = 0
(x - 0.5)² - 1 = 0
(x - 0.5)² = 1

Taking the square root of both sides gives |x - 0.5| = 1.

Therefore, x - 0.5 = 1 or x - 0.5 = -1, leading to the solutions x = 1.5 and x = -0.5.

Example 2: Solve 2x² + 4x + 8 = 0

Divide the entire equation by 2 to simplify: x² + 2x + 4 = 0. Complete the square for x² + 2x:

x² + 2x + 1 - 1 + 4 = 0
(x + 1)² + 3 = 0
(x + 1)² = -3

Since the square of a real number cannot be negative, this equation has no real solutions.

Example 3: Solve x² - 8x + 20 = 0

Complete the square for x² - 8x. Half of -8 is -4, and (-4)² = 16.

x² - 8x + 16 - 16 + 20 = 0
(x - 4)² + 4 = 0
(x - 4)² = -4

Again, no real solution exists. In complex numbers, where i² = -1, we proceed.

x - 4 = ±√(-4)
x - 4 = ±2i

Thus, the complex solutions are x = 4 + 2i and x = 4 - 2i.

Additional Practice Problems

Try solving these equations by completing the square:

x² + 8x - 9 = 0
x² + 4x = 0
x² + 4x + 4 = 0

Solutions

Solutions: x = -9 or x = 1.
Solutions: x = 0 or x = -4.
Solution: x = -2.

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