Location of roots of two quadratic w.r.t. to third quadratic

If $\displaystyle a,b,c \in R$ such that $\displaystyle abc \neq0$ If $\displaystyle x_1$ is a root of $\displaystyle a^2x^2+bx+c=0, x_2$ is a root of $\displaystyle a^2x^2-bx-c=0$ and $\displaystyle x_1 > x_2 >0$ then the equation $\displaystyle a^2x^2+2bx+2c=0$ has roots $\displaystyle x_3$ .

Prove that $\displaystyle x_3$ lies between $\displaystyle x_1 \& x_2$

Let f(x) = $\displaystyle a^2x^2+2bx+2c=0$

$\Rightarrow $\displaystyle f(x_1)=a^2x_1^2+2bx_1+2c=-a^2x_1^2$

$\displaystyle \Rightarrow$$\displaystyle f(x_2) = a^2x_2^2+2bx_2+2c=3a^2x_2^2$

$\displaystyle \Rightarrow$$\displaystyle f(x_1)(x_2) = (3a^2x_2^2)(-a^2x_1^2) <0$

$\displaystyle \Rightarrow$Thus one root of $\displaystyle a^2x^2+2bx+2c=0$ will lie between $\displaystyle x_1 \& x_2$

Please provide explanation on the last statement of this answer how it derived ...Thanks..

Re: Location of roots of two quadratic w.r.t. to third quadratic

Hey sachinrajsharma.

What is the definition of f(x1)(x2)? Is it in terms of f(x1) and g(x2)?

Re: Location of roots of two quadratic w.r.t. to third quadratic

$\displaystyle f(x_{1})f(x_{2})<0$ implies that $\displaystyle f(x_{1}) \text{ and } f(x_{2})$ are opposite in sign.

Since $\displaystyle f(x)$ is continuous, it must be zero at some point between $\displaystyle x_{1}$ and $\displaystyle x_{2}.$