College Algebra (11th Edition)

$\text{Discriminant: } 0 \\\text{Number of solutions: } 1 \\\text{Type of solutions: rational solution }$
$\bf{\text{Solution Outline:}}$ To evaluate the discriminant of the given equation, $x(9x+6)=-1 ,$ 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 $\text{ 2 irrational solutions .}$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} x(9x)+x(6)=-1 \\\\= 9x^2+6x=-1 .\end{array} In the form $ax^2+bx+c=0,$ the given equation is equivalent to \begin{array}{l}\require{cancel} 9x^2+6x+1=0 .\end{array} In the equation above, $a= 9 ,$ $b= 6 ,$ and $c= 1 .$ Using the Discriminant Formula which is given by $b^2-4ac,$ the value of the discriminant is \begin{array}{l}\require{cancel} (6)^2-4(9)(1) \\\\= 36-36 .\end{array} Since the discriminant is $\text{ is equal to zero ,}$ then there is $\text{ 1 rational solutions .}$ Hence, the given equation has the following properties: \begin{array}{l}\require{cancel} \text{Discriminant: } 0 \\\text{Number of solutions: } 1 \\\text{Type of solutions: rational solution } .\end{array}