First slide

Theory of equations

Question

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

Moderate

A

7
B

4
C

1
D

none of these

Solution

Putting $y = | x - 2 |,$ we can rewrite (1) as
$\begin{array}{l} y^{2} + y - 2 = 0 or y^{2} + 2 y - y - 2 = 0 \\ \Rightarrow y (y + 2) - (y + 2) = 0 \Rightarrow (y - 1) (y + 2) = 0 \\ As y = | x - 2 | \geq 0, y + 2 \geq 2 \\ ∴ y - 1 = 0 \Rightarrow y = 1 \\ \Rightarrow | x - 2 | = 1 \Rightarrow x - 2 = \pm 1 \\ \Rightarrow x = 2 \pm 1 \Rightarrow x = 3 or 1 \end{array}$
$∴$ Sum of the roots = 4

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