First slide

Theory of equations

Question

$If a, b, c are three distinct positive real numbers then the number of real roots of a x^{2} + 2 b | x | - c = 0 is$

Moderate

A

0
B

4
C

2
D

None of these

Solution

$|x| = \frac{- 2 b \pm \sqrt{4 b^{2} + 4 a c}}{2 a} a, b, c are positive. So, | x | = - b \pm \sqrt{b^{2} + a c}$
$x = \pm [- b \pm \sqrt{b^{2} + a c}]$
$x = - b + \sqrt{b^{2} + a c}, - b - \sqrt{b^{2} + a c}, b + \sqrt{b^{2} + a c}, b - \sqrt{b^{2} + a c}$
$\Rightarrow x has two real values, neglecting x = - b - \sqrt{b^{2} + a c}, b - \sqrt{b^{2} + a c} as | x | \geq 0$

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