Sums Cubes n' Roots

[nippon276]

Here's my problem: The sum of the cubes of the roots for the equation x²-80x+k=0 is 244,160. Find the larger of the two roots for this equation. Help is appreciated!

 

[nippon276]

You can't get started, like using quadratic formula to solve x^2 - 80x + k = 0 ?

There's second variable k that confuses me.

 

[nippon276]

Never mind everyone! I've solved it on my own!

 

[stapel]

The sum of the cubes of the roots for the equation x²-80x+k=0 is 244,160. Find the larger of the two roots for this equation.

Using the hint provided earlier:

\(\displaystyle x\, =\, \dfrac{-(-80)\, \pm\, \sqrt{(-80)^2\, -\, 4(1)(k)\,}}{2(1)}\, =\, \dfrac{80\, \pm\, \sqrt{6400\, -\, 4k\,}}{2}\, =\, 40\, \pm\, \sqrt{1600\, -\, k\,}\)

So the Quadratic Formula gives the two roots in terms of k. Then:

\(\displaystyle \left(40\, -\, \sqrt{1600\, -\, k\,}\right)^3\, +\, \left(40\, +\, \sqrt{1600\, -\, k\,}\right)^3\, =\, 244,160\)

Solve for k (you should get k = 1116), and then for the larger root (using the "plus" part of the Quadratic Formula). :wink: