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

grislingatb 2021-11-17 Answered
To calculate: The solution of the equation \(\displaystyle\sqrt{{{x}+{4}}}-{2}={x}\).

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Expert Answer

oces3y
Answered 2021-11-18 Author has 7480 answers
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.
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