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Question Number 172006 by Mikenice last updated on 23/Jun/22

solve: (√(x^2 +5x+2)) −(√(x^2 +5 )) =1

$${solve}: \\ $$$$\sqrt{{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{2}}\:−\sqrt{{x}^{\mathrm{2}} +\mathrm{5}\:}\:=\mathrm{1} \\ $$

Answered by Rasheed.Sindhi last updated on 24/Jun/22

(√(x^2 +5x+2)) −(√(x^2 +5 )) =1 (√(x^2 +5x+2)) =1+(√(x^2 +5 )) x^2 +5x+2=1+x^2 +5 +2(√(x^2 +5 )) 5x−3=2(√(x^2 +5 )) 25x^2 −30x+9=4x^2 +20 21x^2 −30x−11=0 x=((30±(√(900+924)))/(42))=((30±4(√(114)))/(42)) =((15±2(√(114)))/(21)) Answers should be tested for validity.

$$\sqrt{{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{2}}\:−\sqrt{{x}^{\mathrm{2}} +\mathrm{5}\:}\:=\mathrm{1} \\ $$$$\sqrt{{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{2}}\:=\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{5}\:}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{2}=\mathrm{1}+{x}^{\mathrm{2}} +\mathrm{5}\:+\mathrm{2}\sqrt{{x}^{\mathrm{2}} +\mathrm{5}\:}\: \\ $$$$\mathrm{5}{x}−\mathrm{3}=\mathrm{2}\sqrt{{x}^{\mathrm{2}} +\mathrm{5}\:}\: \\ $$$$\mathrm{25}{x}^{\mathrm{2}} −\mathrm{30}{x}+\mathrm{9}=\mathrm{4}{x}^{\mathrm{2}} +\mathrm{20} \\ $$$$\mathrm{21}{x}^{\mathrm{2}} −\mathrm{30}{x}−\mathrm{11}=\mathrm{0} \\ $$$${x}=\frac{\mathrm{30}\pm\sqrt{\mathrm{900}+\mathrm{924}}}{\mathrm{42}}=\frac{\mathrm{30}\pm\mathrm{4}\sqrt{\mathrm{114}}}{\mathrm{42}} \\ $$$$\:\:\:\:\:\:=\frac{\mathrm{15}\pm\mathrm{2}\sqrt{\mathrm{114}}}{\mathrm{21}} \\ $$$${Answers}\:{should}\:{be}\:{tested}\:{for}\:{validity}. \\ $$

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