# Applying the Quadratic Formula (Part 1)

38 questions

# Applying the Quadratic Formula (Part 1)

##### Problem 1

1) Select $\textbf{all}$ the equations that have 2 solutions. Write each corresponding letter in the answer box and separate letters with commas.

a) $(x + 3)^{2} = 9 \quad \quad\quad$ b) $(x - 5)^{2} = -5 \quad \quad\quad$ c) $(x + 2)^{2} - 6 = 0 \quad \quad\quad$ d) $(x - 9)^{2} + 25 = 0 \quad \quad$ e) $(x + 10)^{2} = 1$

f) $(x - 8)^{2} = 0 \quad \quad$ g) $5 = (x + 1)(x + 1) \quad \quad$

##### Problem 2

A frog jumps in the air. The height, in inches, of the frog is modeled by the function $h(t) = 60t - 75t^{2}$, where $t$ is the time after it jumped, measured in seconds.

2) Solve $60t - 75t^{2} = 0$.

3) What do the solutions tell us about the jumping frog?

##### Problem 3

A tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeled by the equation $f(t) = 4 + 12t - 16t^{2},$ where $t$ is measured in seconds since the ball was thrown.

4) Find the solutions to the equation $0 = 4 + 12t - 16t^{2}$.

5) What do the solutions tell us about the tennis ball?

##### Problem 4

Rewrite each quadratic expression in standard form.

6) $(x + 1)(7x + 2)$

7) $(8x + 1)(x - 5)$

8) $(2x + 1)(2x - 1)$

9) $(4 + x)(3x - 2)$

##### Problem 5

Find the missing expression in parentheses so that each pair of quadratic expressions is equivalent. Then, show that your expression meets this requirement.

10) $(4x - 1)(\underline{\quad\quad\quad\quad\quad})$ and $16x^{2} - 8x + 1$

11) Show that your expression makes the pair equivalent.

12) $(9x + 2)(\underline{\quad\quad\quad\quad\quad})$ and $9x^{2} - 16x - 4$

13) Show that your expression makes the pair equivalent.

14) $(\underline{\quad\quad\quad\quad\quad})(-x + 5)$ and $-7x^{2} + 36x - 5$

15) Show that your expression makes the pair equivalent.

##### Problem 6

The number of downloads of a song is a function, $f$, of the number of weeks, $w$, since the song was released. The equation $f(w) = 100,000 \cdot \left(\frac{9}{10}\right)^{w}$ defines this function.

16) What does the number 100,000 tell you about the downloads?

17) What about the $\frac{9}{10}$?

18) Is $f(-1)$ meaningful in this situation?

19) Explain your reasoning.

##### Problem 7

Consider the equation $4x^{2} - 4x - 15 = 0$.

Identify the values of $a$, $b$, and $c$ that you would substitute into the quadratic formula to solve the equation.

20) $a = \underline{\quad}$?

21) $b = \underline{\quad}$?

22) $c = \underline{\quad}$?

Evaluate each expression using the values of $a$, $b$, and $c$.

23) $-b$

24) $b^{2}$

25) $4ac$

26) $b^{2} - 4ac$

27) $\sqrt{b^{2} - 4ac}$

28) $-b \pm \sqrt{b^{2} - 4ac}$

29) $2a$

30) $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

31) The solutions to the equation are $x = -\frac{3}{2}$ and $x = \frac{5}{2}$. Do these match the values of the last expression you evaluated in the previous question?

##### Problem 8

Describe the graph of $y = -x^{2}$.

32) Does it open upward or downward?

33) Where is its y-intercept?

34) What about its x-intercept?

Without graphing, describe how adding $16x$ to $-x^{2}$ would change each feature of the graph of $y = -x^{2}$. (If you get stuck, consider writing the expression in factored form.)

35) the x-intercepts

36) the vertex

37) the y-intercept

38) the direction of opening of the U-shape graph