Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.5 - Equations - 1.5 Exercises - Page 57: 99

Answer

$x=3$

Work Step by Step

$2x+\sqrt{x+1}=8$ Take the $2x$ to the right side of the equation: $\sqrt{x+1}=8-2x$ Square both sides of the equation: $(\sqrt{x+1})^{2}=(8-2x)^{2}$ $x+1=64-2(8)(2x)+4x^{2}$ $x+1=64-32x+4x^{2}$ Take all terms to the right side of the equation: $0=64-32x+4x^{2}-x-1$ $64-32x+4x^{2}-x-1=0$ Simplify this equation by combining like terms: $4x^{2}-33x+63=0$ Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For this equation, $a=4$, $b=-33$ and $c=63$ Substitute the known values in the formula: $x=\dfrac{-(-33)\pm\sqrt{(-33)^{2}-4(4)(63)}}{2(4)}=\dfrac{33\pm\sqrt{1089-1008}}{8}=...$ $...=\dfrac{33\pm\sqrt{81}}{8}=\dfrac{33\pm9}{8}$ Our two preliminary solutions are: $x=\dfrac{33+9}{8}=\dfrac{21}{4}$ $x=\dfrac{33-9}{8}=3$ Substitute the solutions in the equation to check them: $2(3)+\sqrt{3+1}=8$ $6+2=8$ $8=8$ True $2\Big(\dfrac{21}{4}\Big)+\sqrt{\dfrac{21}{4}}=8$ $\dfrac{21+\sqrt{21}}{2}\ne8$ False The answer is $x=3$
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