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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. 169 (page 906)

In the following exercises, solve. $2 (x^{2} - 5) - 5 (x^{2} - 5) + 2 = 0$

Short Answer

Expert verified

Solution for equation will be

$x = + \sqrt{7}$
$x = - \sqrt{7}$
$x = + \sqrt{\frac{11}{2}}$
$x = - \sqrt{\frac{11}{2}}$

Step by step solution

01

Step 1. Given Data

Quadratic Equation is $2 (x^{2} - 5) - 5 (x^{2} - 5) + 2 = 0$

02

Step 2. To Find

We have to solve it and find the value of x.

03

Step 3. Explanation

Let the value of $(x^{2} - 5) = u$

Then the equation is -

$2 (x^{2} - 5) - 5 (x^{2} - 5) + 2 = 0$

Put the value of $(x^{2} - 5) = u$

$2 u^{2} - 5 u + 2 = 0$

$(u - 2) (u - \frac{1}{2}) = 0$

By using Zero product property -

$u - 2 = 0$ , $u - \frac{1}{2} = 0$

$u = 2$ , $u = \frac{1}{2}$

04

Step 4. Calculation

Replace u with $(x^{2} - 5)$

$x^{2} - 5 = 2$ , $x^{2} - 5 = \frac{1}{2}$

$x^{2} = 2 + 5$ , $x^{2} = \frac{1}{2} + 5$

$x^{2} = 7$ , $x^{2} = \frac{11}{2}$

$x = 卤 \sqrt{7}$ , $x = 卤 \sqrt{\frac{11}{2}}$

There are four solutions -

$x = + \sqrt{7}$
$x = - \sqrt{7}$
$x = + \sqrt{\frac{11}{2}}$
$x = - \sqrt{\frac{11}{2}}$

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