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Question

If the difference of the roots of the quadratic equation is 5 and the difference of their cubes is 215 , find the quadratic equation.

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Solution

Let a and b be the roots of the quadratic equation .
Therefore, ab=5 .......(1)
And, a3b3=215
We know that,
a3b3=(ab)3+3ab(ab)
215=53+15ab
15ab=90
ab=6
Also, (ab)2=(a+b)24ab
(a+b)2=49
a+b=±7
Therefore, quadratic equation is,
x2(±7)x+6=0
x2±7x+6=0

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