Question

if the difference of the roots of the quadratic equation is 5 and the difference if their cubes is 215, find the quadratic equation ........Plz help me ASAP

Answer 1

$let \: the \: roots \: be = a \: and \: b \\ then \: given \: that \: a - b = 5 and \: {a}^{3 - {b}^{3} } = 215 \\ {a - b}^{3} = {a}^{3} - 3 {a}^{2} b + 3a {b}^{2} - {b}^{3} = {5}^{3} \\ ( {a}^{3} - {b}^{3} ) - 3 {a}^{2} b + 3a {b}^{2} = 125 \\ 215 - 3( {a}^{2} b - {b}^{2} a) = 125 \\ - 3( {a}^{2} b - {b}^{2} a) = 125 - 215 \\ {a}^{2} b - {b}^{2} a = - 90 \div - 3 \\ = {a}^{2} b - {b}^{2} a = 30 \\ so \: the \: eqution \: is \\ {a}^{2} b - {b}^{2} a - 30 = 0$