If $a,b,c \in R$ such that $abc \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$ then the equation $a^2x^2+2bx+2c=0$ has roots $x_3$ .
Prove that $x_3$ lies between $x_1 \& x_2$
Let f(x) = $a^2x^2+2bx+2c=0$
$\Rightarrow f(x_1)=a^2x_1^2+2bx_1+2c=-a^2x_1^2$
$\Rightarrow f(x_2) = a^2x_2^2+2bx_2+2c=3a^2x_2^2$
$\Rightarrow f(x_1)(x_2) = (3a^2x_2^2)(-a^2x_1^2) <0$
$\Rightarrow $Thus one root of $a^2x^2+2bx+2c=0$ will lie between $x_1 \& x_2$
Please provide explanation on the last statement of this answer how it derived …Thanks..
Best Answer
Make a rough sketch of your findings. Will the curve cut the x-axis?