Algebra: A Combined Approach (4th Edition)

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

Chapter 11 - Section 11.3 - Solving Equations by Using Quadratic Methods - Exercise Set: 10

Answer

$x=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$

Work Step by Step

$\dfrac{5}{x-2}+\dfrac{4}{x+2}=1$ Multiply the whole equation by $(x-2)(x+2)$: $(x-2)(x+2)\Big(\dfrac{5}{x-2}+\dfrac{4}{x+2}=1\Big)$ $5(x+2)+4(x-2)=(x-2)(x+2)$ $5x+10+4x-8=x^{2}-4$ Take all terms to the right side of the equation and simplify it: $0=x^{2}-4-5x-10-4x+8$ $x^{2}-9x-6=0$ Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a=1$, $b=-9$ and $c=-6$. Substitute the known values into the formula and simplify: $x=\dfrac{-(-9)\pm\sqrt{(-9)^{2}-4(1)(-6)}}{2(1)}=\dfrac{9\pm\sqrt{81+24}}{2}=...$ $...=\dfrac{9\pm\sqrt{105}}{2}=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$ The original equation is not undefined for neither of the values of $x$ found. Our final answer is $x=\dfrac{9}{2}\pm\dfrac{\sqrt{105}}{2}$
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