Question

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

Answer 1

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(x^2+3x-6)`

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+3x-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:

`ax^2+bx+c=0`

We know:

`x=(-b+-sqrt(b^2-4ac))/(2a)=(-3+-sqrt(3^2-4xx1xx-6))/(2xx1)`
`=(-3+-sqrt(9+24))/(2)=(-3+-sqrt(33))/(2)=(-3+-sqrt(33))/(2)`
`=(-3-5.74)/2` or `(-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`.