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.

A:

\((2+x)(2-x)\)

B:

\((x+9)(x-9)\)

C:

\((2+x)(x-2)\)

D:

\((x+y)(x-y)\)

1:

\(x^2 -4\)

2:

\(81 - x^2\)

3:

\(x^2 - y^2\)

4:

\(4-x^2\)

5:

\(x^2 - 81\)

(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.)