SOLUTION: Given that the equation x(x-2p)=q(x-p) has real roots for all real values of p and q. If q=3, find a non-zero value for p so that the roots are rational.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Given that the equation x(x-2p)=q(x-p) has real roots for all real values of p and q. If q=3, find a non-zero value for p so that the roots are rational.      Log On


   

Question 1072645: Given that the equation x(x-2p)=q(x-p) has real roots for all real values of p and q. If q=3, find a non-zero value for p so that the roots are rational.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Simplifying the quadratic equation from the description:
x%5E2-2px=3x-3p
x%5E2-2px-3x%2B3p
x%5E2-%282p%2B3%29x%2B3p=0


The discriminant must be 0 for rational roots:
%28-1%29%5E2%2A%282p%2B3%29%5E2-4%2A3p=0
4p%5E2-12p%2B9-12p=0
4p%5E2-24p%2B9=0
%284p-3%29%5E2=0
highlight%28p=3%2F4%29

Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!

Given that the equation x(x-2p)=q(x-p) has real roots for all real values of p and q. If q=3, find a non-zero value for p so that the roots are rational.
For the roots to be rational, the discriminant must be = 0, or a positive PERFECT SQUARE integer, such as 1, or 4, or 9, etc.

When the discriminant is set as ≥ 0, the result is the following quadratic inequality: 4p%5E2+%2B+9+%3E=+0

Solving for p gives 2 IMAGINARY numbers.