Question

the sum of two roots of quadratic equation is 5 and sum of their cubes is 35 find the equation

Answer 1

let the two roots be x and y .
so, x+y = 5
and,x^3 +y^3 = 35
=> (x+y)^3 - 3x^2y -3xy^2 = 35
=> (5)^3 - 3xy(x+y) = 35
=> 125-3xy × 5 = 35
=> 125-15xy = 35
=> xy = 90/15 = 6
=>y = 6/x
putting y in x+y = 5, we get,
x+6/x = 5
x^2 +6 =5x
x^2-5x+6 =0
(x-3)(x-2) = 0, by solving the eqn.
so ,x = 3 or x = 2 .so y = 6/3 = 2
now we have the two roots as 3 and 2.
now,
let the equation be ax^2 +bx +c
so,x+y = -b/a
3+2 = -b/a
5. = -b/a
and, x×y = c/a
3×2 = c/ a
6 = c/a
so,a = 1,b = -5 and c= 6.
so the quadratic eqn.is x^2 - 5x +6.