if the zeros of the polynomial ax^2+bx+c be in ratio m:n then prove that b^2mn=(m^2+n^2)ac - Maths - Polynomials

Question

if the zeros of the polynomial ax^2+bx+c be in ratio m:n then prove
that b^2mn=(m^2+n^2)ac

Vijay Kumar Gupta · Accepted Answer

Dear Student,

Let α and β are the roots of the quadratic equation ax2+bx+c=0
Therefore,       sum of roots α+β=ba     product of roots αβ=caIt is given that the roots are in the ratio of m:nTherefore,           α:β=m:n  ⇒    αβ=mn  ⇒    α+βα-β=m+nm-n       by applying Componendo and Dividendo  ⇒    α+β2α-β2=m+n2m-n2       squaring both sides⇒    α+β2α+β2-4αβ=m+n2m-n2⇒    ba2ba2-4ca=m+n2m-n2⇒    b2b2-4ac=m+n2m-n2⇒    b2m-n2=m+n2b2-4ac⇒    b2m-n2=b2m+n2-4acm+n2⇒   4acm+n2= =b2m+n2-b2m-n2⇒   4acm+n2= =b2m+n2-m-n2⇒   4acm+n2= =b24mn⇒   acm+n2= =b2mn⇒   b2mn=acm+n2

Regards