Lesson 16

The Quadratic Formula

The practice problem answers are available at one of our IM Certified Partners

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. \(\text- x^2 + 16x + 64 = 0\)

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 \(x = \frac56 \pm \frac{\sqrt{13}}{6}\).

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:

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

(From Algebra1, 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?
(From Algebra1, 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. \(7x + 14x^2 = 42\)

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.
(From Algebra1, Unit 4, Lesson 14.)