# How do you find the roots, real and imaginary, of y=2x^2 -12x -5 using the quadratic formula?

Dec 7, 2015

There are two real roots that are irrational: $x = 3 \pm \frac{1}{2} \sqrt{46}$

#### Explanation:

The quadratic formula says the roots of the function $y = a {x}^{2} + b x + c$ are $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.

For the present problem, $a = 2$, $b = - 12$, and $c = - 5$. Hence, the roots are

$\frac{12 \pm \sqrt{144 - 4 \cdot 2 \cdot \left(- 5\right)}}{2 \cdot 2} = 3 \pm \frac{1}{4} \sqrt{184}$

$= 3 \pm \frac{1}{4} \sqrt{4} \sqrt{46} = 3 \pm \frac{1}{2} \sqrt{46}$.