Lesson 16

The Quadratic Formula

Problem 1

For each equation, identify the values of \(a\), \(b\), and \(c\) that you would substitute into the quadratic formula to solve the equation.

  1. \(3x^2 + 8x + 4 = 0\)
  2. \(2x^2 - 5x + 2 = 0\)
  3. \(\text- 9x^2 + 13x - 1 = 0\)
  4. \(x^2 + x - 11 = 0\)
  5. \(\text- x^2 + 16x + 64 = 0\)

Solution

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Problem 2

Use the quadratic formula to show that the given solutions are correct.

  1. \(x^2 + 9x + 20 =0\). The solutions are \(x = \text- 4\) and \(x = \text- 5\).
  2. \(x^2 - 10x + 21 = 0\). The solutions are \(x = 3\) and \(x = 7\).
  3. \(3x^2 - 5x + 1 = 0\). The solutions are \(x = \frac56 \pm \frac{\sqrt{13}}{6}\).

Solution

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Problem 3

Select all the equations that are equivalent to \(81x^2+180x-200=100\)

A:

\(81x^2+180x-100=0\)

B:

\(81x^2+180x+100=200\)

C:

\(81x^2+180x+100=400\)

D:

\((9x+10)^2=400\)

E:

\((9x+10)^2=0\)

F:

\((9x-10)^2=10\)

G:

\((9x-10)^2=20\)

Solution

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(From Unit 7, Lesson 14.)

Problem 4

Technology required. Two objects are launched upward. Each function gives the distance from the ground in meters as a function of time, \(t\), in seconds.

Object A: \(f(t)=25+20t-5t^2\)

Object B: \(g(t)=30+10t-5t^2\)

Use graphing technology to graph each function.

  1. Which object reaches the ground first? Explain how you know.
  2. What is the maximum height of each object?

Solution

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(From Unit 6, Lesson 6.)

Problem 5

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

  1. \(x^2 + 9x + 18 = 0\)
  2. \(4x^2 - 3x + 11 = 0\)
  3. \(81 - x + 5x^2 = 0\)
  4. \(\frac45 x^2 + 3x = \frac13 \)
  5. \(121 = x^2\)
  6. \(7x + 14x^2 = 42\)

Solution

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Problem 6

On the same coordinate plane, sketch a graph of each function.

  • Function \(v\), defined by \(v(x) = |x+6|\)
  • Function \(z\), defined by \(z(x)= |x|+9\)
Blank coordinate grid, Origin O. X axis from negative 12 to 9 by 3’s. Vertical axis from negative 3 to 21 by 3’s.

Solution

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(From Unit 4, Lesson 14.)