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Question Number 148746 by mathdanisur last updated on 30/Jul/21

if   x - y = y - z = 4  find   x^2  + z^2  - 2y^2  = ?

$${if}\:\:\:{x}\:-\:{y}\:=\:{y}\:-\:{z}\:=\:\mathrm{4} \\ $$$${find}\:\:\:{x}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:-\:\mathrm{2}{y}^{\mathrm{2}} \:=\:? \\ $$

Answered by Rasheed.Sindhi last updated on 30/Jul/21

if   x - y = y - z = 4     x^2  + z^2  - 2y^2  = ?                                             _(−)   x=y+4 ,z=y−4    x^2  + z^2  - 2y^2  = (y+4)^2 +(y−4)^2 −2y^2   =y^2 +8y+16+y^2 −8y+16−2y^2   =2y^2 +32−2y^2 =32

$${if}\:\:\:{x}\:-\:{y}\:=\:{y}\:-\:{z}\:=\:\mathrm{4} \\ $$$$\underset{−} {\:\:\:{x}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:-\:\mathrm{2}{y}^{\mathrm{2}} \:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} \\ $$$${x}={y}+\mathrm{4}\:,{z}={y}−\mathrm{4} \\ $$$$\:\:{x}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:-\:\mathrm{2}{y}^{\mathrm{2}} \:=\:\left({y}+\mathrm{4}\right)^{\mathrm{2}} +\left({y}−\mathrm{4}\right)^{\mathrm{2}} −\mathrm{2}{y}^{\mathrm{2}} \\ $$$$={y}^{\mathrm{2}} +\cancel{\mathrm{8}{y}}+\mathrm{16}+{y}^{\mathrm{2}} −\cancel{\mathrm{8}{y}}+\mathrm{16}−\mathrm{2}{y}^{\mathrm{2}} \\ $$$$=\cancel{\mathrm{2}{y}^{\mathrm{2}} }+\mathrm{32}−\cancel{\mathrm{2}{y}^{\mathrm{2}} }=\mathrm{32} \\ $$$$ \\ $$

Commented by mathdanisur last updated on 30/Jul/21

Thank You Ser

$${Thank}\:{You}\:{Ser} \\ $$

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