# Applying the Quadratic Formula (Part 2)

ID: lopob-pufah
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2

32 questions

# Applying the Quadratic Formula (Part 2)

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##### Problem 1

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

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

If this equation is written in standard form, ﻿$ax^{2} + bx + c = 0$﻿, what are the values of ﻿$a$﻿, ﻿$b$﻿, and ﻿$c$﻿?

1) ﻿$a = \underline{\quad}$﻿?

2) ﻿$b = \underline{\quad}$﻿?

3) ﻿$c = \underline{\quad}$﻿?

4) Do you agree with either of them?

True or false? Write below.

##### Problem 2

The equation ﻿$h(t) = -16t^{2} + 80t + 64$﻿ represents the height, in feet, of a potato ﻿$t$﻿ 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.

##### Problem 3

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

True or false? Write below.

10) Show how you know.

##### Problem 4

Lin says she can tell that ﻿$25x^{2} + 40x + 16$﻿ and ﻿$49x^{2} - 112x + 64$﻿ 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 ﻿$25x^{2} + 40x + 16$﻿ 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 ﻿$49x^{2} - 112x + 64$﻿ 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 ﻿$25x^{2} + 40x + 16$﻿ in factored form.

Show Work

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

Show Work
##### Problem 5

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

a)

﻿$x$﻿=﻿$\frac{-5\pm\sqrt{17}}{4}$﻿

b)

﻿$x$﻿=﻿$\frac{5\pm\sqrt{17}}{4}$﻿

c)

﻿$x$﻿=﻿$\frac{-5\pm\sqrt{33}}{4}$﻿

d)

﻿$x$﻿=﻿$\frac{5\pm\sqrt{33}}{4}$﻿

##### 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.

﻿$x^{2} + 11x + 24 = 0$﻿

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.

﻿$4x^{2} + 20x + 25 = 0$﻿

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.

﻿$x^{2} + 8x = 5$﻿

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.

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

a)

X

b)

Y

c)

Z

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

a)

X

b)

Y

c)

Z

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

a)

X

b)

Y

c)

Z

##### Problem 8

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

29) What are the x-intercepts of the graph of ﻿$f$﻿?

Show Work

30) Find the coordinates of the vertex of the graph of ﻿$f$﻿.

Show Work

32) Sketch a graph of ﻿$f$﻿.