Lesson 16

The Quadratic Formula

View Student Lesson

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

For access, consult one of our IM Certified Partners.

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

For access, consult one of our IM Certified Partners.

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

For access, consult one of our IM Certified Partners.

(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

For access, consult one of our IM Certified Partners.

(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

For access, consult one of our IM Certified Partners.

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

For access, consult one of our IM Certified Partners.

(From Unit 4, Lesson 14.)