Lesson 12

Completing the Square (Part 1)

Lesson Narrative

Previously, students saw that a squared expression of the form $$(x+n)^2$$ is equivalent to $$x^2 + 2nx + n^2$$. This means that, when written in standard form $$ax^2 + bx +c$$ (where $$a$$ is 1), $$b$$ is equal to $$2n$$ and $$c$$ is equal to $$n^2$$. Here, students begin to reason the other way around. They recognize that if $$x^2 + bx +c$$ is a perfect square, then the value being squared to get $$c$$ is half of $$b$$, or $$\left(\frac {b}{2}\right)^2$$. Students use this insight to build perfect squares, which they then use to solve quadratic equations.

Students learn that if we rearrange and rewrite the expression on one side of a quadratic equation to be a perfect square, that is, if we complete the square, we can find the solutions of the equation.

Rearranging parts of an equation strategically so that it can be solved requires students to make use of structure (MP7). Maintaining the equality of an equation while transforming it prompts students to attend to precision (MP6).

Learning Goals

Teacher Facing

• Comprehend that to “complete the square” is to determine the value of $c$ that will make the expression $x^2+bx+c$ a perfect square.
• Describe (orally and in writing) how to complete the square.
• Solve quadratic equations of the form $x^2+bx+c=0$ by rearranging terms and completing the square.

Student Facing

• Let’s learn a new method for solving quadratic equations.

Student Facing

• I can explain what it means to “complete the square” and describe how to do it.
• I can solve quadratic equations by completing the square and finding square roots.

Building Towards

Glossary Entries

• completing the square

Completing the square in a quadratic expression means transforming it into the form $$a(x+p)^2-q$$, where $$a$$, $$p$$, and $$q$$ are constants.

Completing the square in a quadratic equation means transforming into the form $$a(x+p)^2=q$$.

• perfect square

A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

• rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into $$b$$ equal parts and finding the point that is $$a$$ of them from 0. We can always write a fraction in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are whole numbers, with $$b$$ not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as $$\frac{7}{10}$$.

The numbers $$3$$, $$\text-\frac34$$, and $$6.7$$ are all rational numbers. The numbers $$\pi$$ and $$\text-\sqrt{2}$$ are not rational numbers, because they cannot be written as fractions or their opposites.