Solve the radical equation. \sqrt{x+16}-\sqrt{x-4}=2

sodni3 2021-10-02 Answered
Solve the radical equation.
\(\displaystyle\sqrt{{{x}+{16}}}-\sqrt{{{x}-{4}}}={2}\)

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Expert Answer

Yusuf Keller
Answered 2021-10-03 Author has 25296 answers
Step 1
Solve the radical
\(\displaystyle\sqrt{{{x}+{16}}}-\sqrt{{{x}-{4}}}={2}\)
Step 2
\(\displaystyle\sqrt{{{x}+{16}}}-\sqrt{{{x}-{4}}}={2}\)
\(\displaystyle\Rightarrow\sqrt{{{x}+{16}}}=\sqrt{{{x}-{4}}}+{2}\)
squaring both sides
\(\displaystyle{\left(\sqrt{{{x}+{16}}}\right)}^{{{2}}}={\left(\sqrt{{{x}-{4}}}+{2}\right)}^{{{2}}}\)
Expanding both sides
\(\displaystyle{x}+{16}={x}-{4}+{2}{\left(\sqrt{{{x}-{4}}}\right)}{\left({2}\right)}+{4}\ \ \ {\left[{\left({a}+{b}\right)}^{{{2}}}={a}^{{{2}}}+{2}{a}{b}+{b}^{{{2}}}\right]}\)
\(\displaystyle\Rightarrow{x}+{16}={x}+{4}\sqrt{{{x}-{4}}}\)
\(\displaystyle\Rightarrow{16}={4}\sqrt{{{x}-{4}}}\)
\(\displaystyle\Rightarrow{\frac{{{16}}}{{{4}}}}=\sqrt{{{x}-{4}}}\)
\(\displaystyle\Rightarrow{4}=\sqrt{{{x}-{4}}}\)
squaring both sides
16=x-4
\(\displaystyle\Rightarrow{20}={x}\)
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