Question

How do you use the quadratic formula to find the roots of the equation $x^2-6x-19=0$?

Answer 1

Step 1: Apply the quadratic formula.

The quadratic formula for the quadratic equation $a{x}^2+bx+c=0$ is $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

Compare the equation $x^2-6x-19=0$ with $a{x}^2+bx+c=0$, we get $a=1,b=-6,c=-19$.

Substitute $a=1,b=-6,c=-19$ into the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, we get

$$\begin{gathered} x=\frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4\left(1\right)\left(-19\right)}}{2\left(1\right)} \\ \Rightarrow x=\frac{6\pm \sqrt{36+76}}{2} \\ \Rightarrow x=\frac{6\pm \sqrt{112}}{2} \\ \therefore x=\frac{6\pm 4\sqrt{7}}{2} \end{gathered}$$

Step 2: Find the first root.

If $x=\frac{6+4\sqrt{7}}{2}$

$$\begin{gathered} \Rightarrow x=\frac{2\left(3+2\sqrt{7}\right)}{2} \\ \therefore x=3+2\sqrt{7} \end{gathered}$$

Step 3: Find the second root.

If $x=\frac{6-4\sqrt{7}}{2}$

$$\begin{gathered} \Rightarrow x=\frac{2\left(3-2\sqrt{7}\right)}{2} \\ \therefore x=3-2\sqrt{7} \end{gathered}$$

Hence, the solutions to the given quadratic equation are $x=3+2\sqrt{7}$and $x=3-2\sqrt{7}$.