Question

let a,b,c be real numbers ( a is not zero) alpha and beta be the roots is equation ax^2+bx+ c=0..Express the roots of the equation a^3x^2+ abcx + c^3=0 in terms of alpha and beta

Answer 1

Equation ax2+bx+c=0ax2+bx+c=0 has roots αα and bb

α+β=−ba+b=−ba

αb=caαb=ca

Equation Ax2+Bx+C=0Ax2+Bx+C=0 has roots α3α3and b3b3

CA=α3b3=(αb)3=c3a3CA=α3b3=(αb)3=c3a3

BA=−(α3+b3)=3αb(α+b)−(α+b)3=3(ca)(−ba)−(−ba)3=b3−3abca3BA= −(α3+b3)=3αb(α+b)−(α+b)3=3(ca)(−ba)−(−ba)3=b3−3abca3

A=a3,B=b3−3abc,C=c3A=a3,B=b3−3abc,C=c3

Quadratic equation with roots α3α3 and b3b3:

a3x2+b(b2−3ac)x+c3=0