## Intermediate Algebra (12th Edition)

Published by Pearson

# Chapter 8 - Section 8.1 - The Square Root Property and Completing the Square - 8.1 Exercises - Page 512: 57

#### Answer

$x=\left\{ -2-\sqrt{6},-2+\sqrt{6} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $x^2+4x-2=0 ,$ use first the properties of equality to express the equation in the form $x^2\pm bx=c.$ Once in this form, complete the square by adding $\left( \dfrac{b}{2} \right)^2$ to both sides of the equal sign. Then express the left side as a square of a binomial while simplify the right side. Then take the square root of both sides (Square Root Property) and use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the properties of equality, in the form $x^2+bx=c,$ the given equation is equivalent to \begin{array}{l}\require{cancel} x^2+4x=2 .\end{array} In the equation above, $b= 4 .$ The expression $\left( \dfrac{b}{2} \right)^2,$ evaluates to \begin{array}{l}\require{cancel} \left( \dfrac{4}{2} \right)^2 \\\\= \left( 2 \right)^2 \\\\= 4 .\end{array} Adding the value of $\left( \dfrac{b}{2} \right)^2,$ to both sides of the equation above results to \begin{array}{l}\require{cancel} x^2+4x+4=2+4 \\\\ x^2+4x+4=6 .\end{array} With the left side now a perfect square trinomial, the equation above is equivalent to \begin{array}{l}\require{cancel} (x+2)^2=6 .\end{array} Taking the square root of both sides (Square Root Property), simplifying the radical and then isolating the variable, the equation above is equivalent to \begin{array}{l}\require{cancel} x+2=\pm\sqrt{6} \\\\ x=-2\pm\sqrt{6} .\end{array} The solutions are \begin{array}{l}\require{cancel} x=-2-\sqrt{6} \\\\\text{OR}\\\\ x=-2+\sqrt{6} .\end{array} Hence, $x=\left\{ -2-\sqrt{6},-2+\sqrt{6} \right\} .$

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