Learn on PengienVision, Algebra 2Chapter 2: Quadratic Functions and Equations

Lesson 5: Completing the Square

In this Grade 11 enVision Algebra 2 lesson, students learn to solve quadratic equations by completing the square, including rewriting expressions as perfect square trinomials using the formula x² + bx + (b/2)² = (x + b/2)². The lesson covers solving equations with real and complex solutions, such as those yielding imaginary roots in the form a ± bi√c, and applies the technique to real-world area problems. It builds on students' prior knowledge of perfect square trinomials and square root methods from Chapter 2 of the quadratic functions unit.

Section 1

Complete the Square of x^2 + bx

Property

To complete the square of $x^2 + bx$ :

Step 1. Identify $b$ , the coefficient of $x$ .

Step 2. Find $(\frac{1}{2}b)^2$ , the number to complete the square.

Section 2

Solve $ax^2+bx+c=0$ by Completing the Square

Property

The process of completing the square works best when the leading coefficient is one.
If the $x^2$ term has a coefficient $a$ other than 1, you must first take a preliminary step to make the coefficient equal to one.

To solve an equation of the form $ax^2 + bx + c = 0$ :
Step 1. Divide both sides of the equation by the leading coefficient, $a$ . This gives an equation of the form $x^2 + (b/a)x + (c/a) = 0$ .
Step 2. Proceed with the standard steps for completing the square.

Examples

To solve $2x^2 + 8x - 10 = 0$ , first divide by 2 to get $x^2 + 4x - 5 = 0$ . Then solve by completing the square: $x^2 + 4x = 5 \implies (x+2)^2 = 9 \implies x+2 = \pm 3$ . The solutions are $x=1$ and $x=-5$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Complete the Square of x^2 + bx

Property

To complete the square of $x^2 + bx$ :

Step 1. Identify $b$ , the coefficient of $x$ .

Step 2. Find $(\frac{1}{2}b)^2$ , the number to complete the square.

Section 2

Solve $ax^2+bx+c=0$ by Completing the Square

Property

Examples

To solve $2x^2 + 8x - 10 = 0$ , first divide by 2 to get $x^2 + 4x - 5 = 0$ . Then solve by completing the square: $x^2 + 4x = 5 \implies (x+2)^2 = 9 \implies x+2 = \pm 3$ . The solutions are $x=1$ and $x=-5$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Complete the Square of x^2 + bx

Property

To complete the square of $x^2 + bx$ :

Step 1. Identify $b$ , the coefficient of $x$ .

Step 2. Find $(\frac{1}{2}b)^2$ , the number to complete the square.

Section 2

Solve $ax^2+bx+c=0$ by Completing the Square

Property

Examples

To solve $2x^2 + 8x - 10 = 0$ , first divide by 2 to get $x^2 + 4x - 5 = 0$ . Then solve by completing the square: $x^2 + 4x = 5 \implies (x+2)^2 = 9 \implies x+2 = \pm 3$ . The solutions are $x=1$ and $x=-5$ .