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.

Answer 1

Let a and b be the roots of the quadratic equation .

Therefore, $a-b=5$ .......(1)

And, $a^3-b^3=215$

We know that,

$a^3-b^3=(a-b{)}^3+3ab(a-b)$

$215=5^3+15ab$

$15ab=90$

$ab=6$

Also, $(a-b{)}^2=(a+b{)}^2-4ab$

$(a+b{)}^2=49$

$a+b=\pm 7$

Therefore, quadratic equation is,

$x^2-(\pm 7)x+6=0$

$x^2\pm 7x+6=0$