First slide

Theory of equations

Question

The product of all real roots of the equation $x^{2} - | x | - 6 = 0$ is

Moderate

A

$- 9$
B

6
C

9
D

36

Solution

The given equation is $x^{2} - | x | - 6 = 0$
If $x > 0,$ $\Rightarrow$ equation is $x^{2} - x - 6 = 0$

$\Rightarrow$ $(x - 3) (x + 2) = 0 \Rightarrow x = 3, x = - 2 \Rightarrow x = 3$ $(∵ x > 0)$

If $x < 0$ , $\Rightarrow$ equation is $x^{2} + x - 6 = 0$

$\Rightarrow$ $(x + 3) (x - 2) = 0 \Rightarrow x = - 3, x = 2 \Rightarrow x = - 3$ $(∵ x < 0)$

Hence, product of all possible real roots $= - 9$ .

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