Lesson 15

Quadratic Equations with Irrational Solutions

Problem 1

Solve each equation and write the solutions using \(\pm\) notation.

  1. \(x^2 = 144\)
  2. \(x^2 = 5\)
  3. \(4x^2 = 28\)
  4. \(x^2 = \frac{25}{4}\)
  5. \(2x^2 = 22\)
  6. \(7x^2 = 16\)

Solution

For access, consult one of our IM Certified Partners.

Problem 2

Match each expression to an equivalent expression.

Solution

For access, consult one of our IM Certified Partners.

Problem 3

  1. Is \(\sqrt{4}\) a positive or negative number? Explain your reasoning.
  2. Is \(\sqrt{5}\) a positive or negative number? Explain your reasoning.
  3. Explain the difference between \(\sqrt{9}\) and the solutions to \(x^2 = 9\).

Solution

For access, consult one of our IM Certified Partners.

Problem 4

Technology required. For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close.

\(x^2+10x+8=0\)

\(x^2-4x-11=0\)

 

Solution

For access, consult one of our IM Certified Partners.

Problem 5

Jada is working on solving a quadratic equation, as shown here.

\(\begin{align} p^2-5p&=0\\p(p-5)&=0\\p-5&=0\\p&=5\end{align}\)

She thinks that her solution is correct because substituting 5 for \(p\) in the original expression \(p^2- 5p\) gives \(5^2 - 5(5)\), which is \(25-25\) or 0.

Explain the mistake that Jada made and show the correct solutions.

Solution

For access, consult one of our IM Certified Partners.

Problem 6

Which expression in factored form is equivalent to \(30x^2 +31x+5\)?

A:

\((6x+5)(5x+1)\)

B:

\((5x+5)(6x+1)\)

C:

\((10x+5)(3x+1)\)

D:

\((30x+5)(x+1)\)

Solution

For access, consult one of our IM Certified Partners.

(From Unit 7, Lesson 10.)

Problem 7

Two rocks are launched straight up in the air. The height of Rock A is given by the function \(f\), where \(f(t) = 4 + 30t - 16t^2\). The height of Rock B is given by \(g\), where \(g(t) = 5 +20t - 16t^2\). In both functions, \(t\) is time measured in seconds and height is measured in feet.

  1. Which rock is launched from a higher point?
  2. Which rock is launched with a greater velocity?

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 6.)

Problem 8

  1. Describe how the graph of \(f(x) = |x|\) has to be shifted to match the given graph.
    Function on x y coordinate plane, no grid, origin O. x axis from negative 2 to 7. Y axis from negative 2 to 3. Line starts at 1 comma 3, decreases until 4 comma 0, increases until 7 comma 3.

    ​​​​​

     

  2. Find an equation for the function represented by the graph.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 4, Lesson 14.)