# Lesson 13

Completing the Square (Part 2)

• Let’s solve some harder quadratic equations.

### Problem 1

Add the number that would make the expression a perfect square. Next, write an equivalent expression in factored form.

1. $$x^2 + 3x$$
2. $$x^2 + 0.6x$$
3. $$x^2 - 11x$$
4. $$x^2 - \frac52 x$$
5. $$x^2 + x$$

### Problem 2

Noah is solving the equation $$x^2 + 8x + 15 = 3$$. He begins by rewriting the expression on the left in factored form and writes $$(x+3)(x+5)=3$$. He does not know what to do next.

Noah knows that the solutions are $$x= \text- 2$$ and $$x = \text- 6$$, but is not sure how to get to these values from his equation.

Solve the original equation by completing the square.

### Problem 3

An equation and its solutions are given. Explain or show how to solve the equation by completing the square.

1. $$x^2 + 20x + 50 = 14$$ . The solutions are $$x = \text- 18$$ and $$x = \text- 2$$.
2. $$x^2 + 1.6x = 0.36$$ . The solutions are $$x = \text- 1.8$$ and $$x = 0.2$$.
3. $$x^2 - 5x = \frac{11}{4}$$. The solutions are $$x = \frac{11}{2}$$ and $$x = \frac{\text- 1}{2}$$.

### Problem 4

Solve each equation.

1. $$x^2-0.5x=0.5$$
2. $$x^2+0.8x=0.09$$
3. $$x^2 + \frac{13}{3}x = \frac{56}{36}$$

### Problem 5

Match each quadratic expression given in factored form with an equivalent expression in standard form. One expression in standard form has no match.

(From Unit 7, Lesson 8.)

### Problem 6

Four students solved the equation $$x^2+225=0$$. Their work is shown here. Only one student solved it correctly.

Student A:

\displaystyle \begin{align} x^2 +225&=0\\ x^2&=\text -225\\ x=15 \quad &\text{ or } \quad x= \text- 15\\ \end{align}\\

Student B:

\displaystyle \begin{align} x^2 +225&=0\\ x^2&=\text -225\\ \text{No} &\text{ solutions} \end{align}\\

Student C:

\displaystyle \begin{align} x^2 +225&=0\\ (x-15)(x+15)&=0\\ x=15 \quad \text{ or } \quad x&= \text- 15\\ \end{align}\\

Student D:

\displaystyle \begin{align} x^2 +225&=0\\ x^2&=225\\ x=15 \quad &\text{ or } \quad x= \text- 15\\ \end{align}\\

Determine which student solved the equation correctly. For each of the incorrect solutions, explain the mistake.

(From Unit 7, Lesson 9.)