# To prove the relation between the roots of three quadratic equations

If $$x_1$$ is a root of $$ax^2+bx+c=0$$ , $$x_2$$ is a root of $$-ax^2+bx+c=0$$ where $$0, show that the equation $$ax^2+2bx+2c=0$$ has a root satisfying $$0.

I have been trying this question with different approaches like using vieta's formula, plotting a rough graph of these equations(locating their roots and vertex) and manipulating the given equations to find any relation between $$x_1,x_2,x_3$$ but i fail to prove the above.I hope for an elegant solution to the given problem. Thanks.

Given that $$ax_1^2+bx_1+c==0~~~(1),~~ -ax_2^2+bx_2+c=0,~~(2)$$ Next $$f(x)=ax^2+2bx+2c \implies f(x_1)=ax_1^2+2bx_1+2c~~~(3),~~~f(x_2)=ax_2^2+2bx_2+2c~~~(4)$$ Use (1) in (4) to get $$f(x_1)=-ax_1^2$$ and (2) in (4) to get $$f(x_2)=3ax_2^2$$ So $$f(x_1)f(x_2)=-3a^2x_1^2 x_2^2<0$$ Therefore by IVT there exists $$x_3 \in (x_1,x_2)$$ such that $$f(x_3)=0.$$