Applying the Quadratic Formula (Part 1)

ID: hovij-novak
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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38 questions

Applying the Quadratic Formula (Part 1)

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Problem 1

1) Select all\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(x + 3)^{2} = 9 \quad \quad\quad b) (x5)2=5(x - 5)^{2} = -5 \quad \quad\quad c) (x+2)26=0(x + 2)^{2} - 6 = 0 \quad \quad\quad d) (x9)2+25=0(x - 9)^{2} + 25 = 0 \quad \quad e) (x+10)2=1(x + 10)^{2} = 1

f) (x8)2=0(x - 8)^{2} = 0 \quad \quad g) 5=(x+1)(x+1)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)=60t75t2h(t) = 60t - 75t^{2}, where tt is the time after it jumped, measured in seconds.

2) Solve 60t75t2=060t - 75t^{2} = 0.

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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+12t16t2,f(t) = 4 + 12t - 16t^{2}, where tt is measured in seconds since the ball was thrown.

4) Find the solutions to the equation 0=4+12t16t20 = 4 + 12t - 16t^{2}.

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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)(x + 1)(7x + 2)

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7) (8x+1)(x5)(8x + 1)(x - 5)

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8) (2x+1)(2x1)(2x + 1)(2x - 1)

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9) (4+x)(3x2)(4 + x)(3x - 2)

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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) (4x1)()(4x - 1)(\underline{\quad\quad\quad\quad\quad}) and 16x28x+116x^{2} - 8x + 1

11) Show that your expression makes the pair equivalent.

12) (9x+2)()(9x + 2)(\underline{\quad\quad\quad\quad\quad}) and 9x216x49x^{2} - 16x - 4

13) Show that your expression makes the pair equivalent.

14) ()(x+5)(\underline{\quad\quad\quad\quad\quad})(-x + 5) and 7x2+36x5-7x^{2} + 36x - 5

15) Show that your expression makes the pair equivalent.

Problem 6

The number of downloads of a song is a function, ff, of the number of weeks, ww, since the song was released. The equation f(w)=100,000(910)wf(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 910\frac{9}{10}?

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

True or false? Write below.

19) Explain your reasoning.

Problem 7

Consider the equation 4x24x15=04x^{2} - 4x - 15 = 0.

Identify the values of aa, bb, and cc that you would substitute into the quadratic formula to solve the equation.

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

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

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

Evaluate each expression using the values of aa, bb, and cc.

23) b​​​​-b

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24) b2b^{2}

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25) 4ac4ac

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26) b24ac​​​​​b^{2} - 4ac

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27) b24ac\sqrt{b^{2} - 4ac}

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28) b±b24ac-b \pm \sqrt{b^{2} - 4ac}

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29) 2a2a

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30) b±b24ac2a\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

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31) The solutions to the equation are x=32x = -\frac{3}{2} and x=52x = \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=x2y = -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 16x16x to x2-x^{2} would change each feature of the graph of y=x2y = -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