Question

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

Answer 1

Find the roots:

The quadratic formula for quadratic equation $a{x}^2+bx+c=0$ is $x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

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

$a=1,b=-6,c=-19$.

Substitute these values into the quadratic formula we get,

$$\begin{gathered} x_{1,2}=\frac{-\left(-6\right)\pm \sqrt{{\left(-6\right)}^2-4\left(1\right)\left(-19\right)}}{2\left(1\right)} \\ \Rightarrow x_{1,2}=\frac{6\pm \sqrt{36+76}}{2} \\ \Rightarrow x_{1,2}=\frac{6\pm \sqrt{112}}{2} \\ \Rightarrow x_{1,2}=\frac{6\pm 4\sqrt{7}}{2} \\ \Rightarrow x_{1,2}=\frac{2\left(3\pm 2\sqrt{7}\right)}{2} \\ \therefore x_{1,2}=\left(3\pm 2\sqrt{7}\right) \end{gathered}$$

Therefore, $x_1=3+2\sqrt{7}$ and $x_2=3-2\sqrt{7}$.

Hence, the solutions to the quadratic equation $x^2-6x-19=0$ are $x_1=3+2\sqrt{7}$ and $x_2=3-2\sqrt{7}$.