# Lesson 14

Completing the Square (Part 3)

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

### 14.1: Perfect Squares in Two Forms

Elena says, “$$(x+3)^2$$ can be expanded into $$x^2 + 6x + 9$$. Likewise, $$(2x+3)^2$$ can be expanded into $$4x^2 + 6x + 9$$.”

Find an error in Elena’s statement and correct the error. Show your reasoning.

### 14.2: Perfect in A Different Way

1. Write each expression in standard form:

1. $$(4x+1)^2$$
2. $$(5x-2)^2$$
3. $$(\frac12 x + 7)^2$$
4. $$(3x+n)^2$$
5. $$(kx+m)^2$$
2. Decide if each expression is a perfect square. If so, write an equivalent expression of the form $$(kx+m)^2$$. If not, suggest one change to turn it into a perfect square.

1. $$4x^2 + 12x + 9$$
2. $$4x^2 + 8x+ 25$$

### 14.3: When All the Stars Align

1. Find the value of $$c$$ to make each expression in the left column a perfect square in standard form. Then, write an equivalent expression in the form of squared factors. In the last row, write your own pair of equivalent expressions.
standard form $$(ax^2+bx+c)$$ squared factors $$(kx+m)^2$$
$$100x^2+80x+c$$
$$36x^2-60x+c$$
$$25x^2+40x+c$$
$$0.25x^2-14x+c$$

2. Solve each equation by completing the square:

$$25x^2 + 40x = \text-12$$

$$36x^2 - 60x + 10 = \text-6$$

### 14.4: Putting Stars into Alignment

Here are three methods for solving $$3x^2 + 8x + 5 = 0$$.

Try to make sense of each method.

Method 1:

\displaystyle \begin {align}3x^2 + 8x + 5 &= 0\\ (3x + 5)(x + 1) &= 0 \end{align}

\displaystyle \begin {align} x = \text- \frac53 \quad \text{or} \quad x = \text-1\end {align}

Method 2:

\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\ (3x)^2 + 8(3x) + 15 &= 0\\ U^2 + 8U + 15 &= 0\\ (U+5)(U+3) &= 0 \end{align}
\displaystyle \begin {align} U = \text-5 \quad &\text{or} \quad U = \text-3\\3x = \text-5 \quad &\text{or} \quad 3x = \text-3\\ x = \text- \frac53 \quad &\text{or} \quad x = \text-1 \end{align}

Method 3:

\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\9x^2 + 24x + 16 &= 1\\(3x + 4)^2 &= 1 \end{align}

\displaystyle \begin {align} 3x+4 = 1 \quad & \text{or} \quad 3x+4 = \text-1\\x = \text-1 \quad & \text{or} \quad x = \text- \frac53 \end {align}

Once you understand the methods, use each method at least one time to solve these equations.

1. $$5x^2 + 17x + 6 = 0$$
2. $$6x^2 + 19x = \text-10$$
3. $$8x^2 - 33x + 4 = 0$$
4. $$8x^2 - 26x = \text-21$$
5. $$10x^2 + 37x = 36$$
6. $$12x^2 + 20x - 77=0$$

Find the solutions to $$3x^2 -6x + \frac{9}{4} = 0$$. Explain your reasoning.

### Summary

In earlier lessons, we worked with perfect squares such as $$(x+1)^2$$ and $$(x-5)(x-5)$$. We learned that their equivalent expressions in standard form follow a predictable pattern:

• In general, $$(x+m)^2$$ can be written as $$x^2 + 2mx + m^2$$.
• If a quadratic expression is of the form $$ax^2 + bx + c$$, and the value of $$a$$ is 1, then the value of $$b$$ is $$2m$$, and the value of $$c$$ is $$m^2$$ for some value of $$m$$.

In this lesson, the variable in the factors being squared had a coefficient other than 1, for example $$(3x+1)^2$$ and $$(2x-5)(2x-5)$$. Their equivalent expression in standard form also followed the same pattern we saw earlier.

squared factors standard form
$$(3x+1)^2$$ $$(3x)^2 + 2(3x)(1) + 1^2 \quad \text{or} \quad 9x^2 +6x + 1$$
$$(2x-5)^2$$ $$(2x)^2 + 2(2x)(\text-5) + (\text-5)^2 \quad \text{or} \quad 4x^2 -20x + 25$$

In general, $$(kx+m)^2$$ can be written as:

$$\displaystyle (kx)^2 + 2 (kx)(m) + m^2$$

or

$$\displaystyle k^2 x^2 + 2kmx + m^2$$

If a quadratic expression is of the form $$ax^2 + bx + c$$, then:

• the value of $$a$$ is $$k^2$$
• the value of $$b$$ is $$2km$$
• the value of $$c$$ is $$m^2$$

We can use this pattern to help us complete the square and solve equations when the squared term $$x^2$$ has a coefficient other than 1—for example: $$16x^2 + 40x = 11$$.

What constant term $$c$$ can we add to make the expression on the left of the equal sign a perfect square? And how do we write this expression as squared factors?

• 16 is $$4^2$$, so the squared factors could be $$(4x + m)^2$$.
• 40 is equal to $$2(4m)$$, so $$2(4m) = 40$$ or $$8m=40$$. This means that $$m = 5$$.
• If $$c$$ is $$m^2$$, then $$c = 5^2$$ or $$c=25$$.
• So the expression $$16x^2 + 40x + 25$$ is a perfect square and is equivalent to $$(4x+5)^2$$.

Let’s solve the equation $$16x^2 + 40x = 11$$ by completing the square!

\displaystyle \begin {align} 16x^2 + 40x &= 11\\ 16x^2 + 40x + 25 &= 11 + 25\\ (4x + 5)^2 &=36\\\\4x+5 = 6 \quad &\text {or} \quad 4x+5= \text-6\\ 4x = 1 \quad &\text {or} \quad 4x = \text-11\\ x=\frac14 \quad &\text {or} \quad x = \text- \frac{11}{4} \end {align}.

### 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.