Question

How many roots do the following equations have? -12x^2 -25x +5 +x^3 =0

Select one:
a. 4
b. 2
c. 3
d. 5

Answer 1

The equation has `3` real roots.

Explanation:

First we write the polynomial in standard form:
`x^3-12x^2-25x+5`

Since the polynomial is of degree three, the fundamental theorem of algebra tells us that it has `3` complex roots. Since (non-real) complex roots to functions with real coefficients only come in conjugate pairs, we can deduce that at least one of these `3` complex roots is real.

I presume the question is asking for the number of real roots, so to determine whether the function has `1` or `3` real roots, I will look at the discriminant of the cubic equation.

If you have a cubic equation in the form `ax^3+bx^2+cx+d=0`, the discriminant will be:
`Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2`

If `Delta>0`, we get three distinct roots, if `Delta=0`, we get real roots but one of them will be a multiple root, and finally if `Delta<0`, we will have `1` real root and `2` complex roots.

In our case, we get `Delta=213385`, so we can conclude that we have `3` real roots.