Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 2 - Linear and Quadratic Functions - Section 2.3 Quadratic Functions and Their Zeros - 2.3 Assess Your Understanding - Page 146: 45


Zeros: $\dfrac{\sqrt{5}-1}{4},\dfrac{-\sqrt{5}-1}{4}$ $x$-intercepts: $\dfrac{\sqrt{5}-1}{4},\dfrac{-\sqrt{5}-1}{4}$

Work Step by Step

To find the zeros of a function $f$, solve the equation $f(x)=0$ The zeros of the function are also the $x-$intercepts. Let $f(x)=0$: $$4x^2-1+2x=0$$ Rearranging the equation: $$4x^2+2x-1=0$$ Comparing $4x^2+2x-1=0$ to $ax^2+bx+c=0$ to find $a,b \text{ and } c$ $$\therefore a = 4, b=2 , c =-1$$ Evaluating the discriminant $b^2-4ac$ $$b^2-4ac = (2)^2-4 \times 4 \times -1 = 20$$ The quadratic formula is given by: $$x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$ $$x= \dfrac{-2 \pm \sqrt{20}}{2\times 4}$$ $$x=\dfrac{-2 \pm 2\sqrt{5}}{8}$$ $\therefore x =\dfrac{\sqrt{5}-1}{4}\hspace{20pt} \text{or} \hspace{20pt} x=\dfrac{-\sqrt{5}-1}{4}$
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