Lesson 20
Rational and Irrational Solutions
 Let’s consider the kinds of numbers we get when solving quadratic equations.
Problem 1
Decide whether each number is rational or irrational.
 10
 \(\frac45 \)
 \(\sqrt4 \)
 \(\sqrt{10}\)
 3
 \(\sqrt{\frac{25}{4}}\)
 \(\sqrt{0.6}\)
Problem 2
Here are the solutions to some quadratic equations. Select all solutions that are rational.
\(5 \pm 2\)
\(\sqrt4 \pm 1\)
\(\frac12 \pm 3\)
\(10 \pm \sqrt3\)
\(\pm \sqrt{25} \)
\(1 \pm \sqrt2 \)
Problem 3
Solve each equation. Then, determine if the solutions are rational or irrational.
 \((x+1)^2 = 4\)
 \((x5)^2 = 36\)
 \((x+3)^2 = 11\)
 \((x4)^2 = 6\)
Problem 4
Here is a graph of the equation \(y=81(x3)^24\).

Based on the graph, what are the solutions to the equation \(81(x3)^2=4\)?
 Can you tell whether they are rational or irrational? Explain how you know.
 Solve the equation using a different method and say whether the solutions are rational or irrational. Explain or show your reasoning.
Problem 5
Match each equation to an equivalent equation with a perfect square on one side.
Problem 6
To derive the quadratic formula, we can multiply \(ax^2+bx+c=0\) by an expression so that the coefficient of \(x^2\) is a perfect square and the coefficient of \(x\) is an even number.
 Which expression, \(a\), \(2a\), or \(4a\), would you multiply \(ax^2+bx+c=0\) by to get started deriving the quadratic formula?
 What does the equation \(ax^2+bx+c=0\) look like when you multiply both sides by your answer?
Problem 7
Here is a graph that represents \(y=x^2\).
On the same coordinate plane, sketch and label the graph that represents each equation:
 \(y=\textx^24\)
 \(y=2x^2+4\)
Problem 8
Which quadratic expression is in vertex form?
\(x^26x+8\)
\((x6)^2+3\)
\((x3)(x6)\)
\((8x)x\)
Problem 9
Function \(f\) is defined by the expression \(\frac{5}{x2}\).
 Evaluate \(f(12)\).
 Explain why \(f(2)\) is undefined.
 Give a possible domain for \(f\).