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Question Number 124423 by ZiYangLee last updated on 03/Dec/20

If (√(213−4x^2 ))−2(√(12−x^2 ))=11, find the value of (√(213−4x^2 ))+2(√(12−x^2 )).

$$\mathrm{If}\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }−\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }=\mathrm{11}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }. \\ $$

Commented by ZiYangLee last updated on 03/Dec/20

Answer: 15

$$\mathrm{Answer}:\:\mathrm{15} \\ $$

Answered by Olaf last updated on 03/Dec/20

(√(213−4x^2 ))−2(√(12−x^2 )) = (((213−4x^2 )−4(12−x^2 ))/( (√(213−4x^2 ))+2(√(12−x^2 )))) = ((213−4×12)/( (√(213−4x^2 ))+2(√(12−x^2 )))) = ((165)/( (√(213−4x^2 ))+2(√(12−x^2 )))) = ((15×11)/( (√(213−4x^2 ))+2(√(12−x^2 )))) = 11 ⇒ (√(213−4x^2 ))+2(√(12−x^2 )) = 15

$$\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }−\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }\:= \\ $$$$\frac{\left(\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} \right)−\mathrm{4}\left(\mathrm{12}−{x}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }}\:= \\ $$$$\frac{\mathrm{213}−\mathrm{4}×\mathrm{12}}{\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }}\:= \\ $$$$\frac{\mathrm{165}}{\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }}\:= \\ $$$$\frac{\mathrm{15}×\mathrm{11}}{\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }}\:=\:\mathrm{11} \\ $$$$\Rightarrow\:\sqrt{\mathrm{213}−\mathrm{4}{x}^{\mathrm{2}} }+\mathrm{2}\sqrt{\mathrm{12}−{x}^{\mathrm{2}} }\:=\:\mathrm{15} \\ $$

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