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9-12 Integrated Math 2 Unit 5 Lesson 20

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Unit 5, Lesson 20

Rational and Irrational Solutions

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Integrated Math 2

Practice

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.

\((x+1)^2 = 4\)
\((x-5)^2 = 36\)
\((x+3)^2 = 11\)
\((x-4)^2 = 6\)

Problem 4

Here is a graph of the equation \(y=81(x-3)^2-4\).

According to the graph, what are the solutions to the equation \(81(x-3)^2=4\)?
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

ForLesson 13: Completing the Square (Part 2)

Match each equation to an equivalent equation with a perfect square on one side.

\(x^2 - 9x = \frac12\)
\(x^2 + 6.4x - 8.9 = 0\)
\(x^2 - 5x = 11\)
\(x^2 + 0.1x + 0.0005 = 0\)
\(x^2 - \frac67 x = \frac{1}{49}\)
\(x^2 + 1.21x = 6.28\)

\((x - 2.5)^2 = 17.25\)
\((x - \frac92)^2 = \frac{83}{4}\)
\((x - \frac37 )^2 = \frac{10}{49}\)
\((x + 0.05)^2 = 0.002\)
\((x + 3.2)^2 = 19.14\)
\((x + 0.605)^2 = 6.646025\)

Problem 6

ForLesson 19: Deriving the Quadratic Formula

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

ForLesson 12: Graphing the Standard Form (Part 1)

Here is a graph that represents \(y=x^2\).

On the same coordinate plane, sketch and label the graph that represents each equation:

\(y=\text-x^2-4\)
\(y=2x^2+4\)

<p>A curve in an x y plane, origin O.</p>

Problem 8

ForLesson 15: Vertex Form

Which quadratic expression is in vertex form?

\(x^2-6x+8\)
\((x-6)^2+3\)
\((x-3)(x-6)\)
\((8-x)x\)