Question

Find the condition for which ax²+bx+c=0 will be in the ratio m:n

Answer 1

Hello!
The quadratic equation = ax²+bx+c=0
The ratio of roots =m:n
Let roots be mx, nx
Sum of roots=-b/a =mx+nx

-b/a=(m+n)x

x= -b/a (m+n) ......(1)


Products of roots =c/a=mx*nx
mnx²=c/a
x²=(c)/a(mn)......(2)


substituting (1 ) in (2)

x²=c/a(mn)

{-b/a(m+n)}²= c/a(mn)

(b)²/a²(m+n²)= c/a(mn)

mnb²=c/a *a²(m+n)²

mnb²=(m+n)²ac

hence the relation between the roots and coefficients when the roots are in the ratio m:n is mnb²=(m+n)²ac

∴ mnb²=(m+n)²ac

hope helped!