## College Algebra (11th Edition)

$\bf{\text{Solution Outline:}}$ To evaluate the discriminant of the given equation, $8x^2=-14x-3 ,$ express first the equation in the form $ax^2+bx+c=0.$ Then identify first the value of $a,b,$ and $c.$ 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 $\text{ 1 distinct rational solution .}$ 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 $\text{ 2 irrational solutions .}$ $\bf{\text{Solution Details:}}$ Using the properties of equality, the given equation is equivalent to \begin{array}{l}\require{cancel} 8x^2=-14x-3 \\\\ 8x^2+14x+3=0 .\end{array} In the equation above, $a= 8 ,$ $b= 14 ,$ 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} (14)^2-4(8)(3) \\\\= 196-96 \\\\= 100 .\end{array} Since the discriminant is $\text{ a positive perfect square ,}$ then there are $\text{ 2 rational solutions .}$