# How do you solve y=x^3 + 3x^2 - 6x  using the quadratic formula?

Jul 15, 2017

This cubic has 3 roots: $x = 0 , - 4.37$ or $1.37$

We can solve it be dividing through by $x$ and using the quadratic formula.

#### Explanation:

This is not a quadratic, it's a cubic, so it'll have 1-3 roots rather than 0-2 (think about the shape of a cubic compared to a parabola-shaped quadratic and how each can cut the x-axis).

It's a tricky one, but we can divide through by x and treat it as:

$y = x \left({x}^{2} + 3 x - 6\right)$

One root of this will be $x = 0$: when $x = 0$, then $y = 0$, which is the definition of a root.

The other roots will be when the parenthesis is equal to $0$, so we have:

${x}^{2} + 3 x - 6 = 0$

And hey presto, we have a quadratic to solve! And we know how to do that: the quadratic formula. For a quadratic in the form:

$a {x}^{2} + b x + c = 0$

We know:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a} = \frac{- 3 \pm \sqrt{{3}^{2} - 4 \times 1 \times - 6}}{2 \times 1}$
$= \frac{- 3 \pm \sqrt{9 + 24}}{2} = \frac{- 3 \pm \sqrt{33}}{2} = \frac{- 3 \pm \sqrt{33}}{2}$
$= \frac{- 3 - 5.74}{2}$ or $\frac{- 3 + 5.74}{2}$

Therefore $x = - 4.37$ or $1.37$

Remember that we had the other root, $x = 0$, so the 3 roots all together are $x = 0 , - 4.37$ or $1.37$.