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Find x+y+z, if
2x^2+y^2+2z^2-8x+2y-2xy+2xz-16y+35=0
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MATHS
Let x,y,z∈R such that 2x
2
+y
2
+2z
2
−8x+2y−2xy+2xz−16z+35=0, then determine x+y+z
December 20, 2019
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Brijraj Bhosale
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ANSWER
Given 2x
2
+2y
2
+2z
2
−8x+2y−2xy+2xz−16z+35=0
⇒(x
2
+y
2
−2xy)+(x
2
+z
2
+2xz)+z
2
−16z−8x+2y+35=0
⇒(x−y)
2
+(x+z)
2
+(z
2
−10z+25)−16z−8x+2y+10=0
⇒((x−y)
2
−2(x−y)+1)+((x+z)
2
−6(x+z)+9)+(z
2
−10z+25)=0
⇒(x−y−1)
2
+(x+z−3)
2
+(z−5)
2
=0
⇒x−y−1=0 …..(i)
x+z=3 ….(ii)
z−5=0 …..(iii)
⇒z=5
Substituting z in (ii), x=−2
substituting z, x in (i)
We get y=−3
∴x+y+z=−2−3+5=0
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