2.2.2c Solving Quadratic Equations – Quadratic Formula

The quadratic equation $a{x}^2+bx+c=0$ can be solved by using the quadratic formula

$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Example 
Use the quadratic formula to find the solutions of the following equations.
a. $x^2+5x-<!--\; -\; -->24=0$
b. $x\left(x+4\right)=10$

Answer
(a)
For the equation $x^2+5x-<!--\; -\; -->24=0$
$\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-(5)\pm \sqrt{{(5)}^2-4(1)(-24)}}{2(1)}\\ x=\frac{-5\pm \sqrt{121}}{2}\\ x=-8\text{ or }x=3\end{array}$

(b)
$\begin{array}{l}x\left(x+\text{ 4}\right)=\text{ 10}\\ {\text{x}}^2+4x-10=0\\ \\ a=1,b=4,c=-10\\ \\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-(4)\pm \sqrt{{(4)}^2-4(1)(-10)}}{2(1)}\\ x=\frac{-4\pm \sqrt{56}}{2}\\ x=1.742\text{ or }x=-5.742\end{array}$