# Lesson 15

Quadratic Equations with Irrational Solutions

- Let’s find exact solutions to quadratic equations even if the solutions are irrational.

### Problem 1

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

- \(x^2 = 144\)
- \(x^2 = 5\)
- \(4x^2 = 28\)
- \(x^2 = \frac{25}{4}\)
- \(2x^2 = 22\)
- \(7x^2 = 16\)

### Problem 2

Match each expression to an equivalent expression.

### Problem 3

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

### 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.

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

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

### 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.

### Problem 6

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

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

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

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

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

### 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 after the rocks are launched and height is measured in feet above the ground.

- Which rock is launched from a higher point?
- Which rock is launched with a greater velocity?

### Problem 8

- Describe how the graph of \(f(x) = |x|\) has to be shifted to match the given graph.
- Find an equation for the function represented by the graph.