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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q. 9.9 (page 864)

Solve $5 r^{2} - 2 = 34$

Short Answer

Expert verified

The solutions of the quadratic equation: $r = \frac{6 \sqrt{5}}{5}, r = - \frac{6 \sqrt{5}}{5}$

Step by step solution

01

Step 1. Given information

Consider the quadratic equation

$5 r^{2} - 2 = 34$

The objective is to solve the quadratic equation by using square root property.

If $x^{2} = k$ , then $x = 卤 \sqrt{k}$

Isolate the quadratic term.

$5 r^{2} = 36$

Divide by 5 to make the coefficient 1.

$\frac{5 r^{2}}{5} = \frac{36}{5} r^{2} = \frac{36}{5}$

Use the Square Root Property.

localid="1645369492576" $r = 卤 \sqrt{\frac{36}{5}}$

02

Step 2. Simplify the radical expression.

Rewrite the radical as a fraction of square roots.

$r = 卤 \frac{\sqrt{36}}{\sqrt{5}}$

Rationalize the denominator.

$r = 卤 \frac{\sqrt{36}}{\sqrt{5}} 路 \frac{\sqrt{5}}{\sqrt{5}} r = 卤 \frac{6 \sqrt{5}}{5}$

Rewrite to show two solutions.

$r = \frac{6 \sqrt{5}}{5}, r = - \frac{6 \sqrt{5}}{5}$

03

Step 3. Check the solutions:

Substitute $r = \frac{6 \sqrt{5}}{5}$ in the original equation.

$5 r^{2} - 2 = 34 5 {(\frac{6 \sqrt{5}}{5})}^{2} - 2 \overset{?}{=} 34 5 (\frac{36 路 5}{25}) - 2 \overset{?}{=} 34 36 - 2 \overset{?}{=} 34 34 = 34$

Substitute $x = - \frac{6 \sqrt{5}}{5}$ in the original equation.

$5 r^{2} - 2 = 34 5 {(- \frac{6 \sqrt{5}}{5})}^{2} - 2 \overset{?}{=} 34 5 (\frac{36 路 5}{25}) - 2 \overset{?}{=} 34 36 - 2 \overset{?}{=} 34 34 = 34$

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