# Applying the Quadratic Formula (Part 1)

ID: hovij-novak
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2

38 questions

# Applying the Quadratic Formula (Part 1)

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##### 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 ﻿$60t - 75t^{2} = 0$﻿.

Show Work

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 ﻿$0 = 4 + 12t - 16t^{2}$﻿.

Show Work

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

##### Problem 4

Rewrite each quadratic expression in standard form.

6) ﻿$(x + 1)(7x + 2)$﻿

Show Work

7) ﻿$(8x + 1)(x - 5)$﻿

Show Work

8) ﻿$(2x + 1)(2x - 1)$﻿

Show Work

9) ﻿$(4 + x)(3x - 2)$﻿

Show Work
##### 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.

17) What about the ﻿$\frac{9}{10}$﻿?

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

True or false? Write below.

##### 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) ﻿$​​​​-b$﻿

Show Work

24) ﻿$b^{2}$﻿

Show Work

25) ﻿$4ac$﻿

Show Work

26) ﻿$​​​​​b^{2} - 4ac$﻿

Show Work

27) ﻿$\sqrt{b^{2} - 4ac}$﻿

Show Work

28) ﻿$-b \pm \sqrt{b^{2} - 4ac}$﻿

Show Work

29) ﻿$2a$﻿

Show Work

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

Show Work

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?

True or false? Write below.
##### Problem 8

Describe the graph of ﻿$y = -x^{2}$﻿.

32) Does it open upward or downward?

33) Where is its y-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.)