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: 33

Answer

$r^2-r+\frac{1}{4}=(r-\frac{1}{2})^2$

Work Step by Step

We add the square of half of the coefficent of $r$ so that the result is a perfect square trinomial. Co-efficient of $r=-1$ Half of -1 is $\frac{1}{2}×-1=\frac{-1}{2}$ Square of $\frac{-1}{2}$ is $\frac{-1}{2}×\frac{-1}{2}=\frac{1}{4}$ We add $\frac{1}{4}$ to $r^2-r$ to make it a perfect square trinomial. Hence it becomes $r^2-r+\frac{1}{4}$ Factored form- $r^2-r+\frac{1}{4}$ $= r^2-\frac{1}{2}r-\frac{1}{2}r +\frac{1}{4} $ $=r(r-\frac{1}{2})-\frac{1}{2} (r-\frac{1}{2})$ (Taking the common factors) $=(r-\frac{1}{2}) (r-\frac{1}{2})$ ($(r-\frac{1}{2})$ is taken common from both the terms) $=(r-\frac{1}{2})^2$
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