Solve the equation. Give the solutions as EXACT numbers

Question

I have to following problem: $$q²+3q=40$$.

The anwser is $q=5$ and $q= −8$ can someone give me the steps?

Answer 1

The quick version is to first move everything to one side:

$$q^2+3q-40=0$$

From here, you can factor if you can spot the factors by inspection or if you have difficulty with that you can rely on the quadratic formula. The above is in the form $aq^2+bq+c=0$ where $a=1,~b=3,~c=-40$ which has roots $\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$. Plugging in the values gives

$$\dfrac{-3\pm\sqrt{3^2-4\cdot 1\cdot (-40)}}{2\cdot 1} = \dfrac{-3\pm \sqrt{169}}{2}=\dfrac{-3\pm 13}{2}$$

which gives roots $5$ and $-8$, as expected.