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. Notice that the equation is given in the standard form, so we can now identify the values of $a,$ $b,$ and $c.$ $$\begin{array}{c}{x}^2-5x-3=0\iff 1\cdot x^2+(\text{-}5)x+(\text{-}3)=0\end{array}$$ Now, let's evaluate the discriminant. Since the discriminant is $37,$ the quadratic equation has two real solutions.