If $ \displaystyle \alpha $ and $ \displaystyle \beta $ be the roots of the equation $ \displaystyle x^{2}+px+q=0 $ then the equation whose roots are $ \displaystyle \alpha ^{2}+\alpha \beta\ and\ \beta ^{2}+\alpha \beta $ is

Let the required roots be y1 = a^2 + ab and y2 = b^2 + ab. We can factor these as y1 = a(a+b) and y2 = b(a+b). Using Vieta's formulas on the original equation x^2 + px + q = 0, we know a+b = -p and ab = q. The sum of the new roots is y1 + y2 = (a+b)^2 = p^2, and their product is y1*y2 = ab(a+b)^2 = q*p^2. The new quadratic equation is x^2 - (sum)x + (product) = 0, which becomes x^2 - p^2 x + p^2 q = 0.

• Thanks
• Wrong explanation