When do two quadratic equations have no roots in common?

1. Divide both equations by their respective quadratic coefficients, you will have then 2 equations that look like : [math]x²-Sx+P [/math]. S would be the sum of the roots, and P their product. 2. read more

Divide both equations by their respective quadratic coefficients, you will have then 2 equations that look like : [math]x²-Sx+P [/math]. S would be the sum of the roots, and P their product. 2. read more

Solved examples to find the conditions for one common root or both common roots of quadratic equations: 1. If the equations x^2 + px + q = 0 and x^2 + px + q = 0 have a common root and p ≠ q, then prove that p + q + 1 = 0. Solution: Let α be the common root of x^2 + px + q = 0 and x^2 + px + q = 0. Then, α^2 + pα + q = 0 and α^2 + pα + q = 0. read more

Do note that the discriminant of the first equation is $b^2-4ac$. Since a,b,c are in G.P, we must have $$b^2=ac$$ And we get the discriminant of the first quadratic to be 0. Hence $ax^2+2bx+c=0$ has equal roots. read more