Solving Quadratic Equations using the Quadratic Formula

We want to use the discriminant of the given quadratic equation to determine the number of real solutions. In the Quadratic Formula, $b^2-4ac$ is the discriminant. $$\begin{array}{c}a{x}^2+bx+c=0\iff x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}\end{array}$$ If we just want to know the number of real solutions, and not the solutions themselves, we only need to work with the discriminant. To do so, we will rearrange the equation and identify the values of $a,$ $b,$ and $c.$ $$\begin{array}{c}3x^2-x=8\iff 3x^2+(\text{-}1)x+(\text{-}8)=0\end{array}$$ We see that $a$ $=$ $3,$ $b$ $=$ $\text{-}1,$ and that $c$ $=$ $\text{-}8.$ Finally, let's evaluate the discriminant. Since the discriminant is $97,$ the quadratic equation has two real solutions.