College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 1 - Review Exercises: 51

Answer

$\text{Discriminant: } 76 \\\text{Number of distinct solutions: } 2 \\\text{Type of solutions: irrational solutions }$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To evaluate the discriminant of the given equation, $ -6x^2+2x=-3 ,$ identify first the values of $a,b,$ and $c$ in the quadratic expression $ax^2+bx+c.$ Then use the Discriminant Formula. If the value of the discriminant is less than zero, then there are $\text{ 2 nonreal complex solutions .}$ If the value is $0,$ then there is $\text{ 1 rational solution .}$ If the value of the discriminant is a positive perfect square, then there are $\text{ 2 rational solutions .}$ Finally, if the value of the discriminant is positive but not a perfect square, there are $\text{ 2 irrational solutions .}$ $\bf{\text{Solution Details:}}$ In the form $ax^2+bx+c=0,$ the given equation is equivalent to \begin{array}{l}\require{cancel} -6x^2+2x+3=0 \\\\ -1(-6x^2+2x+3)=-1(0) \\\\ 6x^2-2x-3=0 .\end{array} In the equation above, $a= 6 ,$ $b= -2 ,$ and $c= -3 .$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is \begin{array}{l}\require{cancel} (-2)^2-4(6)(-3) \\\\= 4+72 \\\\= 76 .\end{array} Since the discriminant is $\text{ is greater than zero and not a perfect square ,}$ then there are $\text{ 2 irrational solutions .}$ Hence, the given equation has the following properties: \begin{array}{l}\require{cancel} \text{Discriminant: } 76 \\\text{Number of distinct solutions: } 2 \\\text{Type of solutions: irrational solutions } .\end{array}
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