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If-z-1-2-x-1-2-y-1-2-Prove-that-x-y-z-2-4xy-




Question Number 109396 by ZiYangLee last updated on 23/Aug/20
If z^(1/2) =x^(1/2) +y^(1/2)   Prove that (x+y−z)^2 =4xy
$$\mathrm{If}\:{z}^{\frac{\mathrm{1}}{\mathrm{2}}} ={x}^{\frac{\mathrm{1}}{\mathrm{2}}} +{y}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\left({x}+{y}−{z}\right)^{\mathrm{2}} =\mathrm{4}{xy} \\ $$
Answered by bemath last updated on 23/Aug/20
z = x+y+2(√(xy))   2(√(xy)) = z−(x+y)  (2(√(xy)))^2  = (z−(x+y))^2 =((x+y)−z)^2   4xy = (x+y−z)^2
$${z}\:=\:{x}+{y}+\mathrm{2}\sqrt{{xy}}\: \\ $$$$\mathrm{2}\sqrt{{xy}}\:=\:{z}−\left({x}+{y}\right) \\ $$$$\left(\mathrm{2}\sqrt{{xy}}\right)^{\mathrm{2}} \:=\:\left({z}−\left({x}+{y}\right)\right)^{\mathrm{2}} =\left(\left({x}+{y}\right)−{z}\right)^{\mathrm{2}} \\ $$$$\mathrm{4}{xy}\:=\:\left({x}+{y}−{z}\right)^{\mathrm{2}} \\ $$
Commented by Rasheed.Sindhi last updated on 23/Aug/20
∈×⊂∈∣∣∈∩⊤!
$$\in×\subset\in\mid\mid\in\cap\top! \\ $$
Commented by ZiYangLee last updated on 23/Aug/20
Yay!
$$\mathrm{Yay}! \\ $$

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