# How do you find the roots, real and imaginary, of y= (3x+14)x+30x-x^2+14  using the quadratic formula?

May 26, 2018

See explanation.

#### Explanation:

To calculate the roots we first have to transform the function into $y = a {x}^{2} + b x + c$.

$y = \left(3 x + 14\right) x + 30 x - {x}^{2} + 14$

$y = 3 {x}^{2} + 14 x + 30 x - {x}^{2} + 14$

$y = 2 {x}^{2} + 30 x + 14$

Now we can use the quadratic formula:

First calculate the determinant:

$\Delta = {30}^{2} - 4 \cdot 2 \cdot 14 = 900 - 122 = 788$

The determinant is positive, so the function has two different real roots:

${x}_{1} = \frac{- b - \sqrt{\Delta}}{2 a} = \frac{- 30 - \sqrt{788}}{4}$

${x}_{1} = \frac{- b + \sqrt{\Delta}}{2 a} = \frac{- 30 + \sqrt{788}}{4}$