Question

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Q. If the difference of roots of the quadratic equation x² + 4px + p = 0 is same as the difference of roots of quadratic equation x² + 4qx + q = 0, then find the relation between p and q if p ≠ q.

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Answer 1

p=-q will be your answer

Answer 2

Let the roots of equation x²+4px+p=0 be α1 and β1
Let the roots of equation x²+4qx+q=0 be α2 and β2
the difference of roots of both equations is same
⇒α1-β1=α2-β2
but (α-β)=√(b²-4ac)/a
⇒√((4p)²-4p)=√((4q)²-4q)
⇒((4p)²-4p)=((4q)²-4q)
⇒16p²-4p=16q²-4q
⇒4p(4p-1)=4q(4q-1)
⇒p(4p-1)=q(4q-1)
⇒4p²-p=4q²-q
⇒4p²-4q²=p-q
⇒4(p²-q²)=p-q
⇒4=p-q/p²-q²
⇒4=1/p+q
⇒4(p+q)=1
⇒4p+4q=1
⇒4p=1-4q