Question

In the following exercises, solve by using the Quadratic Formula.

$r^2-r-42=0$

Short answer

The solutions are$r=-6andr=7.$

Step-by-step solution

Given information.

The given information is$r^2-r-42=0$

Use the quadratic formula.

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

Now we compare the given equation with the standard form we get $a=1,b=-1andc=-42.$

Now we substitute these values in the quadratic formula.

$$\begin{gathered} r=\frac{-\left(-1\right)\pm \sqrt{{\left(-1\right)}^2-4\left(1\right)\left(-42\right)}}{2\left(1\right)} \\ r=\frac{1\pm \sqrt{1+168}}{2} \\ r=\frac{1\pm \sqrt{169}}{2} \\ r=\frac{1\pm 13}{2} \end{gathered}$$

The two equations arelocalid="1653588225244" $r=\frac{1+13}{2}andr=\frac{1-13}{2}.$

Solve both the equations for r.

$$\begin{gathered} r=\frac{1+13}{2} \\ r=\frac{14}{2} \\ r=7 \\ Or \\ r=\frac{1-13}{2} \\ r=\frac{-12}{2} \\ r=-6 \end{gathered}$$

Therefore, the two solutions are$r=-6andr=7.$