• Apr 27th 2013, 09:43 AM
sachinrajsharma
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$Please provide explanation on the last statement of this answer how it derived ...Thanks.. • Apr 27th 2013, 04:43 PM chiro Re: Location of roots of two quadratic w.r.t. to third quadratic Hey sachinrajsharma. What is the definition of f(x1)(x2)? Is it in terms of f(x1) and g(x2)? • Apr 28th 2013, 12:13 AM BobP Re: Location of roots of two quadratic w.r.t. to third quadratic$\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}.\$