If $a,b,c \in R$ such that $c \neq0$ 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 $x_1 > x_2 >0$.....

Question

Problem :

If $a,b,c \in R$ such that $c \neq0$ 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 $x_3$ such that ( which of the following statement is correct in this context)

(a) $x_1 > x_3 > x_2$

(b) $x_3 > x_1 > x_2$

(c) $x_1 > x_2 > x_3$

If $x_1$ is a root of the given equation then : f($x_1)$ = $a^2x_1^2+bx_1+c=0,$

Also if $x_2$ is root of equation then : $a^2x_2^2-bx_2-c=0 $

Please suggest how to take it further... thanks...

Answer 1

Let $g(x) = a^2 x^2 + 2bx + 2c$

Observe that $$g(x_1) = a^2 x_1^2 + 2bx_1 + 2c = -a^2 x_1^2 < 0,$$

$$ g(x_2) = a^2 x_2^2 + 2bx_2 + 2c = 3 a^2 x_2^2 > 0$$

Can you apply the Intermediate value theorem?

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Note, since the leading coefficient of $g(x)$ is positive, hence $g(\infty) = \infty$, which means that the second root is in the range $[x_1, \infty)$. Hence both roots are positive.