First slide

Theory of equations

Question

Number of real roots of the equation $\sqrt{x} + \sqrt{x - \sqrt{(1 - x)}} = 1$ is

Moderate

A

0
B

1
C

2
D

3

Solution

We have,
$\begin{array}{ll} \sqrt{x} + \sqrt{x - \sqrt{(1 - x)}} & = 1 \\ \Rightarrow \sqrt{x - \sqrt{1 - x}} & = 1 - \sqrt{x} \end{array}$
On squaring both sides, we get
$\begin{aligned} x - \sqrt{1 - x} & = 1 + x - 2 \sqrt{x} \\ \Rightarrow & - \sqrt{1 - x} & = 1 - 2 \sqrt{x} \end{aligned}$
Again, squaring on both sides, we get
$\begin{aligned} 1 - x & = 1 + 4 x - 4 \sqrt{x} \\ 4 \sqrt{x} & = 5 x \\ \Rightarrow \sqrt{x} & = \frac{4}{5} [on squaring both sides] \\ \Rightarrow x & = \frac{16}{25} \end{aligned}$
Hence, the number of real solutions is 1.

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