Algebra: A Combined Approach (4th Edition)

Published by Pearson
ISBN 10: 0321726391
ISBN 13: 978-0-32172-639-1

Chapter 11 - Section 11.1 - Solving Quadratic Equations by Completing the Square - Exercise Set: 39

Answer

$x=(\frac{-1+\sqrt 5}{2}, \frac{-1-\sqrt 5}{2})$

Work Step by Step

Step-1 : Add $1$ on both sides of the equation $x^2+x=1$ Step -2 : Add the square of half of the co-efficient of $x$ to both sides. Co-efficient of $x =1$ Half of $1 = \frac{1}{2}×1=\frac{1}{2}$ Square of $\frac{1}{2}$ is $\frac{1}{2}×\frac{1}{2}=\frac{1}{4}$ The equation becomes $x^2+x+\frac{1}{4}=1+\frac{1}{4}$ Step-3 Factor the trinomial and simplify the right hand side. $(x+\frac{1}{2})^2=\frac{4+1}{4}$ $(x+\frac{1}{2})^2=\frac{5}{4}$ Step-4 Use the square root property and solve for $x$ $(x+\frac{1}{2})=±\sqrt\frac{5}{4} $ Step-5 Subtract $\frac{1}{2}$ on both the sides $x=-\frac{1}{2}±\sqrt \frac{5}{4}$ Step-6 Simplify the right hand side $x=-\frac{1}{2}±\frac{\sqrt 5}{2}$ $x=\frac{-1±\sqrt 5}{2}$ Therefore the solution set is $(\frac{-1+\sqrt 5}{2}, \frac{-1-\sqrt 5}{2})$
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