Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C $2$

$\sqrt{4-x}+\sqrt{x+9}=5$
For the square root to be defined
$4-x\ge 0\Rightarrow 4\ge xx+9\ge 0\Rightarrow x\ge -9$
$\Rightarrow -9\le x\le 4\cdots \left(1\right)$
Now,
$\sqrt{x+9}=5-\sqrt{4-x}$
Squaring on both the sides, we get
$\Rightarrow x+9=25-10\sqrt{4-x}+4-x$
$\Rightarrow 10\sqrt{4-x}=20-2x\Rightarrow 5\sqrt{4-x}=10-x$
Squaring on both the sides, we get
$\Rightarrow 25(4-x)=100-20x+x^2\Rightarrow x^2+5x=0$
$\Rightarrow x(x+5)=0\Rightarrow x=0,-5$

As $x\in [-9,4]$, so there are $2$ values of $x$ satifying the equation.