Applying the Quadratic Formula (Part 2)

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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32 questions

Applying the Quadratic Formula (Part 2)

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Problem 1

Mai and Jada are solving the equation 2x27x=152x^{2} - 7x = 15 using the quadratic formula but found different solutions.

Mai wrote:Jada wrote:x=7±724(2)(15)2(2)x=7±49(120)4x=7±1694x=7±134x=5orx=32x=(7)±724(2)(15)2(2)x=7±49(120)4x=7±714\begin{array}{cc} \text{Mai wrote:} & \text{Jada wrote:} \\ \begin{array}{l} \\[-1em] x = \frac{- 7 \pm \sqrt{7^{2} - 4(2)(-15)}}{2(2)} \\[-1em] \\ \\[-1em] x = \frac{- 7 \pm \sqrt{49 - (- 120)}}{4} \\[-1em] \\ \\[-1em] x = \frac{- 7 \pm \sqrt{169}}{4} \\[-1em] \\ \\[-1em] x = \frac{- 7 \pm 13}{4} \\[-1em] \\ \\[-1em] x = -5 \qquad \text{or} \qquad x = \frac{3}{2} \\[-1em] \end{array} & \begin{array}{l} \\[-1em] x = \frac{-(-7) \pm \sqrt{-7^{2} - 4(2)(-15)}}{2(2)} \\[-1em] \\ \\[-1em] x = \frac{7 \pm \sqrt{-49 - (-120)}}{4} \\[-1em] \\ \\[-1em] x = \frac{7 \pm \sqrt{71}}{4} \\[-1em] \end{array} \end{array}

If this equation is written in standard form, ax2+bx+c=0ax^{2} + bx + c = 0, what are the values of aa, bb, and cc?

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

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

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

4) Do you agree with either of them?

True or false? Write below.

5) Explain your reasoning.

Problem 2

The equation h(t)=16t2+80t+64h(t) = -16t^{2} + 80t + 64 represents the height, in feet, of a potato tt seconds after it was launched from a mechanical device.

6) Write an equation that would allow us to find the time the potato hits the ground.

7) Solve the equation without graphing.

8) Show your reasoning.

Problem 3

9) Priya found x=3x = 3 and x=1x = -1 as solutions to 3x26x9=03x^{2} - 6x - 9 = 0. Is she correct?

True or false? Write below.

10) Show how you know.

Problem 4

Lin says she can tell that 25x2+40x+1625x^{2} + 40x + 16 and 49x2112x+6449x^{2} - 112x + 64 are perfect squares because each expression has the following characteristics, which she saw in other perfect squares in standard form:

\cdot The first term is a perfect square. The last term is also a perfect square.

\cdot If we multiply a square root of the first term and a square root of the last term and then double the product, the result is the middle term.

Show that 25x2+40x+1625x^{2} + 40x + 16 has the characteristics Lin described.

11) The first term is a perfect square.

12) The last term is also a perfect square.

13) If we multiply a square root of the first term and a square root of the last term and then double the product, the result is the middle term.

Show that 49x2112x+6449x^{2} - 112x + 64 has the characteristics Lin described.

14) The first term is a perfect square.

15) The last term is also a perfect square.

16) If we multiply a square root of the first term and a square root of the last term and then double the product, the result is the middle term.

Write each expression in factored form.

17) Write 25x2+40x+1625x^{2} + 40x + 16 in factored form.

Show Work

18) Write 49x2112x+6449x^{2} - 112x + 64 in factored form.

Show Work
Problem 5

19) What are the solutions to the equation 2x25x1=02x^{2} - 5x - 1 = 0?

a)

xx=5±174\frac{-5\pm\sqrt{17}}{4}

b)

xx=5±174\frac{5\pm\sqrt{17}}{4}

c)

xx=5±334\frac{-5\pm\sqrt{33}}{4}

d)

xx=5±334\frac{5\pm\sqrt{33}}{4}

Problem 6

Solve each equation by rewriting the quadratic expression in factored form and using the zero product property, or by completing the square. Then, check if your solutions are correct by using the quadratic formula.

x2+11x+24=0x^{2} + 11x + 24 = 0

20) Solve the equation by rewriting the quadratic expression in factored form and using the zero product property, or by completing the square.

Show Work

21) Check if your solutions are correct by using the quadratic formula.

4x2+20x+25=04x^{2} + 20x + 25 = 0

22) Solve the equation by rewriting the quadratic expression in factored form and using the zero product property, or by completing the square.

Show Work

23) Check if your solutions are correct by using the quadratic formula.

x2+8x=5x^{2} + 8x = 5

24) Solve the equation by rewriting the quadratic expression in factored form and using the zero product property, or by completing the square.

Show Work

25) Check if your solutions are correct by using the quadratic formula.

Problem 7

Here are the graphs of three equations.

Match each graph with the appropriate equation.

A template for answering this question. Ask your instructor for an alternative.

26) y = 10(23)xy \ = \ 10\left(\frac{2}{3}\right)^x

a)

X

b)

Y

c)

Z

27) y = 10(14)xy \ = \ 10\left(\frac{1}{4}\right)^x

a)

X

b)

Y

c)

Z

28) y = 10(35)xy \ = \ 10\left(\frac{3}{5}\right)^x

a)

X

b)

Y

c)

Z

Problem 8

The function f is defined by f(x)=(x+1)(x+6)f(x) = (x + 1)(x + 6).

29) What are the x-intercepts of the graph of ff?

Show Work

30) Find the coordinates of the vertex of the graph of ff.

Show Work

31) Show your reasoning.

32) Sketch a graph of ff.

A template for answering this question. Ask your instructor for an alternative.