Solving Quadratic Equations using the Quadratic Formula

We want to use the discriminant of the given quadratic equation to determine the number and type of 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 and type of solutions, and not the solutions themselves, we only need to work with the discriminant. Let's first write all the terms on the left-hand side. $$\begin{array}{c}{x}^2+7x=\text{-}11\iff x^2+7x+11=0\end{array}$$ Having rewritten the equation, we can now identify the values of $a,$ $b,$ and $c.$ $$\begin{array}{c}1x^2+7x+11=0\end{array}$$ Finally, let's evaluate the discriminant. Since the discriminant is $5,$ greater than zero, the quadratic equation has two real solutions.