First slide

Theory of equations

Question

The number of solutions of $\sqrt{4 - x} + \sqrt{x + 9} = 5$ is

Moderate

A

0
B

1
C

2
D

3

Solution

Note that $4 - x \geq 0$ and $x + 9 \geq 0 \Rightarrow - 9 \leq x \leq 4$
We can write (1) as $\sqrt{x + 9} = 5 - \sqrt{4 - x}$
Squaring both the sides we get
$\begin{array}{l} x + 9 = 25 - 10 \sqrt{4 - x} + 4 - x \\ \Rightarrow 10 \sqrt{4 - x} = 20 - 2 x \Rightarrow 5 \sqrt{4 - x} = 10 - x \end{array}$
Squaring both the sides we get
$\begin{array}{l} 25 (4 - x) = 100 - 20 x + x^{2} \Rightarrow x^{2} + 5 x = 0 \\ \Rightarrow x (x + 5) = 0 \Rightarrow x = 0 or x = - 5 \end{array}$
Both $x = 0$ and $x = - 5$ satisfy (1).

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