How do you find the roots, real and imaginary, of y= 12x^2 + 35x + 72  using the quadratic formula?

Feb 13, 2018

$x = - \frac{35}{24} \pm \frac{\sqrt{2231}}{24} i$

Explanation:

The equation:

$y = 12 {x}^{2} + 35 x + 72$

is a quadratic in standard form:

$y = a {x}^{2} + b x + c$

with $a = 12$, $b = 35$ and $c = 72$

It has zeros given by the quadratic formula:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$\textcolor{w h i t e}{x} = \frac{- \left(\textcolor{b l u e}{35}\right) \pm \sqrt{{\left(\textcolor{b l u e}{35}\right)}^{2} - 4 \left(\textcolor{b l u e}{12}\right) \left(\textcolor{b l u e}{72}\right)}}{2 \left(\textcolor{b l u e}{12}\right)}$

$\textcolor{w h i t e}{x} = \frac{- 35 \pm \sqrt{1225 - 3456}}{24}$

$\textcolor{w h i t e}{x} = \frac{- 35 \pm \sqrt{- 2231}}{24}$

$\textcolor{w h i t e}{x} = - \frac{35}{24} \pm \frac{\sqrt{2231}}{24} i$

Note that $2231 = 23 \cdot 97$ has no square factors, so $\sqrt{2231}$ cannot be simplified.