First slide

Theory of equations

Question

The number of solutions of
$\sqrt{x + 1} - \sqrt{x - 1} = 1 (x \in R)$
is

Moderate

A

1
B

2
C

4
D

infinite

Solution

For the equation to make sense we must have
$x + 1 \geq 0$ and $x - 1 \geq 0 \Rightarrow x \geq - 1, x \geq 1$ .i.e. $x \geq 1$
We rewrite equation as
$\sqrt{x + 1} = 1 + \sqrt{x - 1}$
and square both the sides to obtain
$x + 1 = 1 + x - 1 + 2 \sqrt{x - 1}$
$\Rightarrow \frac{1}{2} = \sqrt{x - 1} \Rightarrow \frac{1}{4} = x - 1$ or $x = \frac{5}{4}$
Also, $x = 5 / 4$ satisfies the given equation.

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