Integers are the best!

Algebra Level 3

$\large p(x)=x^4-5x^2+4$ $\large q(x)=7x^2+5x-2$

$p(x)=0$ has two positive integer roots out of it's four roots as $\alpha$ and $\beta$ such that $\alpha<\beta$ .

$q(x)=0$ has an integer root $\gamma$ .

Then which one of the following is true?

This is an original problem and belongs to the set My Creations

\alpha+\beta=\gamma

None of these

\alpha-\beta=\gamma

\alpha+\beta+\gamma=0

1 solution

Skanda Prasad
Nov 10, 2017

I'll give a hint. That might help...

Though it can be simplified easily, there's a better way.

HINT : For a Quadratic Equation $ax^2+bx+c=0$ ,

$\rightarrow$ If $a+b+c=0$ , then the roots are $1$ and $\dfrac{c}{a}$ are the roots.

$\rightarrow$ If $a-b+c=0$ , then $-1$ and $\dfrac{-c}{a}$ are the roots.