Question

solve the equation for all real solutions in simplest form.
$r^2 - r - 14 = 2r$
answer attempt 1 out of 2
additional solution no solution
$r = $
submit ans

Explanation:

Step1: Rearrange to standard quadratic form

Subtract $2r$ from both sides:
$r^2 - r - 14 - 2r = 0$
$r^2 - 3r - 14 = 0$

Step2: Identify quadratic coefficients

For $ax^2+bx+c=0$, here $a=1$, $b=-3$, $c=-14$.

Step3: Apply quadratic formula

Use $r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$:
$r=\frac{-(-3)\pm\sqrt{(-3)^2-4(1)(-14)}}{2(1)}$

Step4: Simplify discriminant and solve

Calculate discriminant: $\sqrt{9 + 56}=\sqrt{65}$
$r=\frac{3\pm\sqrt{65}}{2}$

Answer:

$r=\frac{3+\sqrt{65}}{2}$ and $r=\frac{3-\sqrt{65}}{2}$