Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 8 - Section 8.2 - The Quadratic Formula - 8.2 Exercises: 52

Answer

$a=9$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To find the value of $ a $ such that the given equation, $ at^2+24t+16=0 ,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable. $\bf{\text{Solution Details:}}$ In the equation above, $a= a ,$ $b= 24 ,$ and $c= 16 .$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is \begin{array}{l}\require{cancel} (24)^2-4(a)(16) \\\\= 576-64a .\end{array} Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then \begin{array}{l}\require{cancel} 576-64a=0 .\end{array} Using the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} -64a=-576 \\\\ a=\dfrac{-576}{-64} \\\\ a=9 .\end{array} Hence, the given equation has $1$ rational solution when $ a=9 .$
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