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Question Number 21459 by x² - y²@gmail.com last updated on 24/Sep/17

Solve (√(x + 1 − 4(√(x − 3)))) + (√(x + 22 + 10(√(x − 3)))) = 7

$$\mathrm{Solve} \\ $$$$\sqrt{{x}\:+\:\mathrm{1}\:−\:\mathrm{4}\sqrt{{x}\:−\:\mathrm{3}}}\:+\:\sqrt{{x}\:+\:\mathrm{22}\:+\:\mathrm{10}\sqrt{{x}\:−\:\mathrm{3}}}\:=\:\mathrm{7} \\ $$

Answered by Tikufly last updated on 27/Sep/17

Sol: Put x−3=a^2 , then the equation becomes (√(a^2 +4−4a )) + (√(a^2 +25+10a)) = 7 ⇒(√((a−2)^2 )) + (√((a+5)^2 ))= 7 ⇒(a−2)+(a+5)=7 ⇒2a+3=7 ⇒a = 2 ⇒(√(x−3)) =2 ⇒x−3 = 4 ⇒x = 7

$$ \\ $$$$\mathrm{Sol}:\:\mathrm{Put}\:{x}−\mathrm{3}={a}^{\mathrm{2}} ,\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{becomes} \\ $$$$\sqrt{{a}^{\mathrm{2}} +\mathrm{4}−\mathrm{4}{a}\:}\:+\:\sqrt{{a}^{\mathrm{2}} +\mathrm{25}+\mathrm{10}{a}}\:=\:\mathrm{7} \\ $$$$\Rightarrow\sqrt{\left({a}−\mathrm{2}\right)^{\mathrm{2}} }\:+\:\sqrt{\left({a}+\mathrm{5}\right)^{\mathrm{2}} \:}=\:\mathrm{7} \\ $$$$\Rightarrow\left({a}−\mathrm{2}\right)+\left({a}+\mathrm{5}\right)=\mathrm{7} \\ $$$$\Rightarrow\mathrm{2}{a}+\mathrm{3}=\mathrm{7} \\ $$$$\Rightarrow{a}\:=\:\mathrm{2} \\ $$$$\Rightarrow\sqrt{{x}−\mathrm{3}}\:=\mathrm{2} \\ $$$$\Rightarrow{x}−\mathrm{3}\:=\:\mathrm{4} \\ $$$$\Rightarrow{x}\:=\:\mathrm{7} \\ $$$$ \\ $$

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