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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.456. (page 984)

Solve the quadratic equation.
$x^{4} + 4 x^{2} - 32 = 0$

Short Answer

Expert verified

The solutions are $x = 卤 2$ and $x = 卤 2 \sqrt{2} i$ .

Step by step solution

01

Step 1. Given information.

The given equation is $x^{4} + 4 x^{2} - 32 = 0$ .

02

Step 2. Simplify the equation by factorization.

$\begin{array}{rcl} x^{4} + 4 x^{2} - 32 & = & 0 \\ {(x^{2})}^{2} + 4 x^{2} - 32 & = & 0 \\ {(x^{2})}^{2} + 8 x^{2} - 4 x^{2} - 32 & = & 0 \\ x^{2} (x^{2} + 8) - 4 (x^{2} + 8) & = & 0 \\ (x^{2} + 8) (x^{2} - 4) & = & 0 \end{array}$

03

Step 3. Apply the zero product property and solve for x.

$\begin{array}{rcl} x^{2} + 8 & = & 0 \\ x^{2} & = & - 8 \\ x & = & 卤 \sqrt{- 8} \\ = & 卤 \sqrt{8} \sqrt{- 1} \\ = & 卤 2 \sqrt{2} i 鈰� 鈰� 鈰� (i = \sqrt{- 1}) \end{array}$ and $\begin{array}{rcl} x^{2} - 4 & = & 0 \\ x^{2} & = & 4 \\ x & = & 卤 \sqrt{4} \\ = & 卤 2 \end{array}$

04

Step 4. Simplified answer.

Hence, the required solutions are $x = 卤 2$ and $x = 卤 2 \sqrt{2} i$ .

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