Prove that $\frac{a}{2}x^2+bx+c=0$ has a root between $x_1$ and $x_2$.

Question

If $x_1$ and $x_2$ are non-zero roots of the equations $ax^2+bx+c=0$ and $-ax^2+bx+c=0$

respectively. Prove that $\frac{a}{2}x^2+bx+c=0$ has a root between $x_1$ and $x_2$.

Please help me ..

Answer 1

Let $f(x)=\frac{a}{2}x^2+bx+c$, then $f(x_1)=-\frac{a}{2}x_1^2$ and $f(x_2)=\frac{3a}{2}x_2^2$. Consequently $f(x_1)f(x_2)<0$. By continuity it should have a root in $(x_1,x_2)$.