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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.9.18 (page 867)

Solve ${(2 t - 8)}^{2} = - 10$

Short Answer

Expert verified

The solutions of the quadratic equation $t = 4 + i \sqrt{\frac{5}{2}}, t = 4 - i \sqrt{\frac{5}{2}}$

Step by step solution

01

Step 1. Given information

Consider the quadratic equation

${(2 t - 8)}^{2} = - 10$

Use the Square Root Property and simplify the radical.

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

$(2 t - 8) = 卤 \sqrt{- 10} (2 t - 8) = 卤 \sqrt{10} i$

Add 8 to both sides.

$2 t = 卤 \sqrt{10} i + 8$

Divide both sides by 2.

localid="1645460079112" $t = \frac{卤 \sqrt{10} i + 8}{2}$

Rewrite in standard form.

localid="1645460187619" $t = \frac{卤 \sqrt{10} i}{2} + \frac{8}{2} t = \frac{卤 \sqrt{5 路 2} i}{2} + 4 t = 卤 \sqrt{\frac{5}{2}} i + 4$

Rewrite to show two solutions.

localid="1645460289551" $t = \sqrt{\frac{5}{2}} i + 4, t = - \sqrt{\frac{5}{2}} i + 4$

02

Step 2. Check for solutions.

Substitute $t = \sqrt{\frac{5}{2}} i + 4$ in the original equation.

${(2 t - 8)}^{2} = - 10 {(2 (\sqrt{\frac{5}{2}} i + 4) - 8)}^{2} \overset{?}{=} - 10 - 2 - 8 \overset{?}{=} - 10 - 10 = - 10$

Substitute $t = - \sqrt{\frac{5}{2}} i + 4$ in the original equation.

${(2 t - 8)}^{2} = - 10 {(2 (- \sqrt{\frac{5}{2}} i + 4) - 8)}^{2} \overset{?}{=} - 10 - 2 - 8 \overset{?}{=} - 10 - 10 = - 10$

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