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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. 160 (page 906)

In the following exercises, solve. $4 x^{4} - 5 x^{2} + 1 = 0$

Short Answer

Expert verified

Solution for equation will be

$x = + 1$
$x = - 1$
$x = + \frac{1}{2}$
$x = - \frac{1}{2}$

Step by step solution

01

Step 1. Given Data

Quadratic Equation is $4 x^{4} - 5 x^{2} + 1 = 0$

02

Step 2. To Find

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

03

Step 3. Explanation

Let the value of $x^{2} = u$

Then the equation is -

$4 x^{4} - 5 x^{2} + 1 = 0$

$4 {(x^{2})}^{2} - 5 x^{2} + 1 = 0$

Put the value of $x^{2} = u$

$4 u^{2} - 5 u + 1 = 0$

$(u - 1) (u - \frac{1}{4}) = 0$

By using Zero product property -

$u - 1 = 0$ , role="math" localid="1645393991904" $u - \frac{1}{4} = 0$

$u = 1$ , $u = \frac{1}{4}$

04

Step 4. Calculation

Replace u with $x^{2}$

$x^{2} = 1$ , $x^{2} = \frac{1}{4}$

Using the square root property -

role="math" localid="1645394220329" $x = 卤 \sqrt{1}$ , $x = 卤 \sqrt{\frac{1}{4}}$

$x = 卤 1$ , $x = 卤 \frac{1}{2}$

There are four solutions -

$x = + 1$
$x = - 1$
$x = + \frac{1}{2}$
$x = - \frac{1}{2}$

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