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

1 Answer
David G.
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(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.

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