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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. 9.10 (page 864)

Solve $3 t^{2} + 6 = 70$

Short Answer

Expert verified

The solutions are $t = \frac{8 \sqrt{3}}{3}, t = - \frac{8 \sqrt{3}}{3}$

Step by step solution

01

Step 1. Given information

Consider the quadratic equation

$3 t^{2} + 6 = 70$

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.

$3 t^{2} = 64$

Divide by 3 to make the coefficient 1.

$\frac{3 t^{2}}{3} = \frac{64}{3} t^{2} = \frac{64}{3}$

Use the Square Root Property.

$t = 卤 \sqrt{\frac{64}{3}}$

02

Step 2. Simplify the radical expression.

Rewrite the radical as a fraction of square roots.

$t = 卤 \frac{\sqrt{64}}{\sqrt{3}}$

Rationalize the denominator.

$t = 卤 \frac{\sqrt{64}}{\sqrt{3}} 路 \frac{\sqrt{3}}{\sqrt{3}} t = 卤 \frac{8 \sqrt{3}}{3}$

Rewrite to show two solutions.

$t = \frac{8 \sqrt{3}}{3}, t = - \frac{8 \sqrt{3}}{3}$

03

Step 3. Check the solutions:

Substitute $t = \frac{8 \sqrt{3}}{3}$ in the original equation.

role="math" localid="1645373895892" $3 t^{2} + 6 = 70 3 {(\frac{8 \sqrt{3}}{3})}^{2} + 6 \overset{?}{=} 70 3 (\frac{64 路 3}{9}) + 6 \overset{?}{=} 70 64 + 6 \overset{?}{=} 70 70 = 70$

Substitute $t = \frac{8 \sqrt{3}}{3}$ in the original equation.

$3 t^{2} + 6 = 70 3 {(- \frac{8 \sqrt{3}}{3})}^{2} + 6 \overset{?}{=} 70 3 (\frac{64 路 3}{9}) + 6 \overset{?}{=} 70 64 + 6 \overset{?}{=} 70 70 = 70$

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