Intermediate Algebra: Connecting Concepts through Application

Published by Brooks Cole
ISBN 10: 0-53449-636-9
ISBN 13: 978-0-53449-636-4

Chapter 4 - Quadratic Functions - 4.6 Solving Quadratic Equations by Using the Quadratic Formula - 4.6 Exercises - Page 372: 7


$\color{blue}{\left\{-\frac{1}{2}, 5\right\}}$

Work Step by Step

The solutions of the quadratic equation $ax^2+bx+c=0$ can be found using the quadratic formula: $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$ Write the given equation in $ax^2+bx+c=0$ form to obtain: $$2x^3-9c-5=0$$ The equation above has $a=2, b=-9, \text{ and } c= -5$. Substitute these values into the quadratic formula to obtain: \begin{align*} x&=\frac{-(-9)\pm\sqrt{(-9)^2-4(2)(-5)}}{2(2)}\\\\ x&=\frac{9\pm \sqrt{81+40}}{4}\\\\ x&=\frac{9\pm \sqrt{121}}{4}\\\\ \end{align*} Thus, $x_1=\dfrac{9+11}{4}=\dfrac{20}{4}=5\\\\$ $x_2=\dfrac{9-11}{4}=\dfrac{-2}{4}=-\dfrac{1}{2}$ Therefore, the solution set is $\color{blue}{\left\{-\frac{1}{2}, 5\right\}}$.
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