# To calculate: The solution of the equation \sqrt{x+4}-2=x.

To calculate: The solution of the equation $$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$.

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Calculation:
Consider the equation, $$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$$\displaystyle\sqrt{{{x}+{4}}}={x}+{2}$$
Now, squaring both sides,
$$\displaystyle{\left(\sqrt{{{x}+{4}}}\right)}^{{{2}}}={\left({x}+{2}\right)}^{{{2}}}$$
$$\displaystyle{x}+{4}={\left({x}\right)}^{{{2}}}+{\left({2}\right)}^{{{2}}}+{2}{\left({x}\right)}{\left({2}\right)}$$
$$\displaystyle{x}+{4}={x}^{{{2}}}+{4}{x}+{4}$$
Further simplify and taking x common from left side of the equation,
$$\displaystyle{x}^{{{2}}}+{3}{x}={0}$$
$$\displaystyle{x}{\left({x}+{3}\right)}={0}$$
Now, apply zero product rule,
$$\displaystyle{x}{\left({x}+{3}\right)}={0}$$
$$\displaystyle{x}={0}$$ or $$\displaystyle{\left({x}+{3}\right)}={0}$$
$$\displaystyle{x}={0}$$ or $$\displaystyle{x}=-{3}$$
Hence, the solution of the equation $$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$ and $$\displaystyle{x}={0}$$ or $$\displaystyle{x}=-{3}$$.
Now, to check the solution put the values of x in the original equation,
Substitute $$\displaystyle{x}={0}$$ in the equation $$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$\sqrt{\left(0\right)+4}-2\left(0\right)$
$2-20$
$$\displaystyle{0}={0}$$
Since, left side is equal to right side so the solution is true.
Now, substitute $$\displaystyle{x}=-{3}$$ in the equation $$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$$\displaystyle\sqrt{{{x}+{4}}}-{2}={x}$$
$\sqrt{\left(-3\right)+4}-2\left(-3\right)$
$1-2-3$
$$\displaystyle-{1}=-{3}$$
Here, left side is not equal to right side so the solution is false.
Therefore, the solution set is (0) and the value $$\displaystyle{x}=-{3}$$ does not check.