# Lesson 14

Completing the Square (Part 3)

### Lesson Narrative

Prior to this lesson, students have solved quadratic equations by completing the square, but all the equations were monic quadratic equations, in which the squared term has a coefficient of 1. In this lesson, students complete the square to solve non-monic quadratic equations, in which the squared term has a coefficient other than 1.

Students begin by noticing that the structure for expanding expressions such as $$(x+m)^2$$ can also be used to expand expressions such as $$(kx+m)^2$$. The expanded expression is always $$k^2x^2 + 2kmx + m^2$$. If the perfect square in standard form is $$ax^2 +bx+c$$, then $$a$$ is $$k^2$$, $$b$$ is $$2km$$, and $$c$$ is $$m^2$$. Recognizing this structure allows students to complete the square for expressions $$ax^2 +bx+c$$ when $$a$$ is not 1, and then to solve equations with such expressions (MP7).

Completing the square when $$a$$ is not 1 can be rather laborious, even when $$a$$ is a perfect square and $$b$$ is an even number. It is even more time consuming and complicated when $$a$$ is not a perfect square and $$b$$ is not an even number. Students are not expected to master the skill of solving non-monic quadratic equations by completing the square. In fact, they should see that this method has its limits and seek a more efficient strategy.

This lesson aims only to show that non-monic quadratic equations can be solved by completing the square and exposing students to how it can be done. This exposure provides some background knowledge that will be helpful when students derive the quadratic formula later.

### Learning Goals

Teacher Facing

• Generalize (orally) a process for completing the square to express any quadratic equation in the form $(kx+m)^2=q$.
• Solve quadratic equations in which the squared term has a coefficient other than 1 by completing the square.

### Student Facing

• Let’s complete the square for some more complicated expressions.

### Student Facing

• I can complete the square for quadratic expressions of the form $ax^2+bx+c$ when $a$ is not 1 and explain the process.
• I can solve quadratic equations in which the squared term coefficient is not 1 by completing the square.

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.

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