# Completing the Square (Part 2)

26 questions

# Completing the Square (Part 2)

##### Problem 1

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

1) Add a number to $x^{2} + 3x$ in order to write it as a perfect square.

2) Write an equivalent expression in factored form.

3) Add a number to $x^{2} + 0.6x$ in order to write it as a perfect square.

4) Write an equivalent expression in factored form.

5) Add a number to $x^{2} - 11x$ in order to write it as a perfect square.

6) Write an equivalent expression in factored form.

7) Add a number to $x^{2} - \frac{5}{2}x$ in order to write it as a perfect square.

8) Write an equivalent expression in factored form.

9) Add a number to $x^{2} + x$ in order to write it as a perfect square.

10) Write an equivalent expression in factored form.

##### Problem 2

11) 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 $x = -2$ and $x = -6$, 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.

12) $x^{2} + 20x + 50 = 14$. The solutions are $x = -18$ and $x = -2$.

13) $x^{2} + 1.6x = 0.36$. The solutions are $x = -1.8$ and $x = 0.2$.

14) $x^{2} - 5x = \frac{11}{4}$. The solutions are $x = \frac{11}{2}$ and $x = \frac{-1}{2}$.

##### Problem 4

Solve each equation.

15) $x^{2} - 0.5x = 0.5$

16) $x^{2} + 0.8x = 0.09$

17) $x^{2} + \frac{13}{3}x = \frac{56}{36}$

##### 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.

18) $(2 \ + \ x)(2 \ - \ x)$

$x^{2}$ - 4

81 - $x^{2}$

$x^{2}$ - $y^{2}$

4 - $x^{2}$

$x^{2}$ - 81

19) $(x \ + \ 9)(x \ - \ 9)$

$x^{2}$ - 4

81 - $x^{2}$

$x^{2}$ - $y^{2}$

4 - $x^{2}$

$x^{2}$ - 81

20) $(2 \ + \ x)(x \ - \ 2)$

$x^{2}$ - 4

81 - $x^{2}$

$x^{2}$ - $y^{2}$

4 - $x^{2}$

$x^{2}$ - 81

21) $(x \ + \ y)(x \ - \ y)$

$x^{2}$ - 4

81 - $x^{2}$

$x^{2}$ - $y^{2}$

4 - $x^{2}$

$x^{2}$ - 81

##### Problem 6

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

Student A:

$\begin{array}{rcl} \\[-1em] x^2 + 225 &=& 0 \\[-1em] \\ \\[-1em] x^2 &=& -225 \\[-1em] \\ \\[-1em] x = 15 & \text{or} & x = -15 \end{array}$

Student B:

$\begin{array}{c} \begin{array}{rcl} \\[-1em] x^2 + 225 &=& 0 \\[-1em] \\ \\[-1em] x^2 &=& -225 \\[-1em] \\ \end{array} \\ \text{No Solutions} \end{array}$

Student C:

$\begin{array}{rcl} \\[-1em] x^2 + 225 &=& 0 \\[-1em] \\ \\[-1em] (x - 15)(x + 15) &=& 0 \\[-1em] \\ \\[-1em] x = 15 &\text{or}& x = -15 \\[-1em] \end{array}$

Student D:

$\begin{array}{rcl} \\[-1em] x^2 + 225 &=& 0 \\[-1em] \\ \\[-1em] x^2 &=& 225 \\[-1em] \\ \\[-1em] x = 15 &\text{or}& x = -15 \\[-1em] \end{array}$

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

22) Which student solved the equation correctly?

23) If Student A was incorrect, explain Student A's mistake.

24) If Student B was incorrect, explain Student B's mistake.

25) If Student C was incorrect, explain Student C's mistake.

26) If Student D was incorrect, explain Student D's mistake.