College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 1 - Review Exercises: 63

Answer

$x=\left\{ -\dfrac{1}{2},\dfrac{1}{2},-i, i \right\}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ 4x^4+3x^2-1=0 ,$ use factoring. Then equate each factor zero and solve the variable using the Square Root Principle. $\bf{\text{Solution Details:}}$ Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the factored form of the equation above is \begin{array}{l}\require{cancel} (4x^2-1)(x^2+1)=0 .\end{array} Equating each factor to zero (Zero Product Property) results to \begin{array}{l}\require{cancel} 4x^2-1=0 \\\\\text{OR}\\\\ x^2+1=0 .\end{array} Isolating the squared variable results to \begin{array}{l}\require{cancel} 4x^2-1=0 \\\\ 4x^2=1 \\\\ x^2=\dfrac{1}{4} \\\\\text{OR}\\\\ x^2+1=0 \\\\ x^2=-1 .\end{array} Taking the square root of both sides (Square Root Principle) results to \begin{array}{l}\require{cancel} x^2=\dfrac{1}{4} \\\\ x=\pm\sqrt{\dfrac{1}{4}} \\\\ x=\pm\sqrt{\left( \dfrac{1}{2} \right)^2} \\\\ x=\pm\dfrac{1}{2} \\\\\text{OR}\\\\ x=\pm\sqrt{-1} \\\\ x=\pm i \text{ (Note that $i=\sqrt{-1}$) } .\end{array} Hence, the solutions are $ x=\left\{ -\dfrac{1}{2},\dfrac{1}{2},-i, i \right\} .$
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