Mastering Completing the Square

• Algebra II
• 11th,12th Grade

Mastering Completing the Square

Learning Objective

Understand how to complete the square and apply it to solve quadratic equations.

What is Completing the Square?

Completing the square is a method used to solve quadratic equations by creating a perfect square trinomial.

Step 1: Write the Equation

Start with a quadratic equation in the form ax^2 + bx + c = 0. Make sure the coefficient of x^2 is 1. If the coefficient of x^2 is not 1 factor it out. 

Step 2: Rewrite the Expression

Step 2. Rewrite the Expression

ax^2 + bx + c

=a(x^2+b/a x)+c

Step 3: Add AND Subtract (b/2a)^2

Add and subtract (b/2a)^2 to the equation to create a perfect square trinomial.

=a(x^2+b/a x)+c

=a(x^2+b/a x+(b/2a)^2-(b/2a)^2)+c

Step 4: Simplify

Write the perfect square trinomial as a squared binomial and solve for x.

=a(x^2+b/a x+(b/2a)^2-(b/2a)^2)+c

=a((x+b/2a)^2-(b/2a)^2)+c

Step 5: Write in Vertex Form

Write the perfect square trinomial as a squared binomial and solve for x.

=a((x+b/2a)^2-(b/2a)^2)+c

=a((x+b/2a)^2-b/4a+c

Example Problem

Given the equation x^2 + 6x +8 , demonstrate step-by-step how to complete the square and solve for x.

Step 2: Plug in the given information from the equation above A=1      B=6      C=8

=a(x^2+b/a x)+c

=1(x^2+6/1 x)+8

Example Problem

Given the equation x^2 + 6x +8 , demonstrate step-by-step how to complete the square and solve for x.

Step 3: Add AND Subtract  (b/2a)^2 using the information given. 

(b/2a)^2=(6/2(1))^2=9

=1(x^2+6x+9-9)+8

Example Problem

Given the equation x^2 + 6x +8 , demonstrate step-by-step how to complete the square and solve for x.

Step 4: Simplify by the finding the factors of (x^2+6x+9) using the box method . 

=1(x^2+6x+9-9)+8

=1((x+3)^2-9)+8

Example Problem

Given the equation x^2 + 6x +8 , demonstrate step-by-step how to complete the square and solve for x.

Step 5: Write in Vertex Form. 

=1((x+3)^2-9)+8 <- Distribute the 1

=(x+3)^2-9+8 <-Combine like terms

=(x+3)^2-1 <-Answer

Practice Problem

Solve the equation x^2 - 8x + 12  using the completing the square method.

Summary

Completing the square is a powerful method for solving quadratic equations and understanding it is essential for success in Algebra 2.

Write down 3 things you learned in this lesson.

Have students enter three things they learned in this lesson. With this they can indicate their own learning efficiency of this lesson.

Write down 2 things you want to know more about.

Here, students enter two things they would like to know more about. This not only increases involvement, but also gives them more ownership.

Ask 1 question about something you haven't quite understood yet.

The students indicate here (in question form) with which part of the material they still have difficulty. For the teacher, this not only provides insight into the extent to which the students understand/master the material, but also a good starting point for the next lesson.