Question

If alpha beta are the roots of a^2 +bx + c = 0. find the values of (beta/alpha^2)+ (alpha/beta^2)

Answer 1

**Ahhh I'm really sorry about this, I'm so careless! Really, I should be the one recalling how to factor sum of cubes lol. :p
This is the last.

I think you mean ax^2 +bx + c = 0.
You can further transform this equation in such a way that the numerical coefficient of x^2 is 1. That is, by dividing all terms by a:
x^2 + (b/a)x + c/a = 0
In this form, recall that the sum of roots is equal to negative of the numerical coeff of x, and the product of roots is equal to the constant:
α + β = -b/a
αβ = c/a
Now, we're looking for the value of (β/α^2) + (α/β^2). We can actually combine this expression to have
(α^3 + β^3) / (α^2*β^2)
which is also (recall how to factor sum of cubes):
(α + β)(α^2 - αβ + β^2) / (α^2*β^2)
We don't have to get the exact values of alpha & beta; we'll just manipulate the equations for their sum & product to arrive the required expression.
Squaring the product equation:
(α^2 * β^2) = c^2 / a^2
Squaring the sum equation:
(α^2 + 2αβ + β^2) = b^2 / a^2
Note that we need only (α^2 - αβ + β^2), so we subtract a 3αβ expression to both sides:
(α^2 + 2αβ + β^2) - 3αβ = (b^2 / a^2) - 3αβ
(α^2 - αβ + β^2) = (b^2 / a^2) - 3c/a
Substituting them,
= (-b/a)*((b^2 / a^2) - 3c/a) / (c^2/a^2)

Simplifying this expression, we have
-b(b^2 - 3ac)/(ac^2)

Ah. I'm really sorry. Hope it doesn't confuse you.
I hope someone would delete my 2 posts above. :3