College Algebra (11th Edition)

$\text{Discriminant: } -188 \\\text{Number of distinct solutions: } 2 \\\text{Type of solutions: nonreal complex solutions }$
$\bf{\text{Solution Outline:}}$ To evaluate the discriminant of the given equation, $8x^2=-2x-6 ,$ 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 $\text{ 2 distinct nonreal complex solutions .}$ 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 $\text{ 2 distinct rational solutions .}$ Finally, if the value of the discriminant is positive but not a perfect square, there are $\text{ 2 distinct irrational solutions .}$ $\bf{\text{Solution Details:}}$ In the form $ax^2+bx+c=0,$ the given equation is equivalent to \begin{array}{l}\require{cancel} 8x^2+2x+6=0 .\end{array} In the equation above, $a= 8 ,$ $b= 2 ,$ and $c= 6 .$ 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(8)(6) \\\\= 4-192 \\\\= -188 .\end{array} Since the discriminant is $\text{ is less than zero ,}$ then there are $\text{ 2 distinct nonreal complex solutions .}$ Hence, the given equation has the following properties: \begin{array}{l}\require{cancel} \text{Discriminant: } -188 \\\text{Number of distinct solutions: } 2 \\\text{Type of solutions: nonreal complex solutions } .\end{array}