Question

if a2+b2+c2-ab-bc-ca = 0 then prove that a=b=c.

Answer 1

HEYA!

ANSWER

a² + b² + c² – ab – bc – ca = 0

Multiply both sides with 2, we get

2( a² + b² + c² – ab – bc – ca) = 0

⇒ 2a² + 2b² + 2c² – 2ab – 2bc – 2ca = 0

⇒ (a² – 2ab + b²) + (b² – 2bc + c²) + (c² – 2ca + a²) = 0

⇒ (a –b)² + (b – c)² + (c – a)² = 0

Since the sum of square is zero then each term should be zero

⇒ (a –b)² = 0,  (b – c)² = 0, (c – a)² = 0

⇒ (a –b) = 0,  (b – c) = 0, (c – a) = 0

⇒ a = b,  b = c, c = a

∴ a = b = c.

$HOPE THIS HELPS$

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Answer 2

Step-by-step explanation:

Hey there !!!!!

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a²+b²+c²-ab-bc-ca=0

a²+b²+c²=ab+bc+ca---Equation 1

Multiplying equation 1 with 2 on both sides

2(a²+b²+c²)=2(ab+bc+ca)

2a²+2b²+2c²=2ab+2bc+2ca

a²+a²+b²+b²+c²+c²=2ab+2bc+2ca

Arranging like terms in form of X²-2XY+Y²=(X-Y)²

(a²-2ab+b²)+(b²-2bc+c²)+(c²-2ac+a²)=0

(a-b)²+(b-c)²+(c-a)²=0----Equation 2

Now square of any number is always positive.But according to equation 2 sum of squares is zero.This is possible only when each term is "0".

So,

(a-b)²=0       (b-c)²=0         (c-a)²=0

⇒⇒a=b       ⇒⇒b=c        ⇒⇒c=a

From above  a=b=c

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Hope this helped you....................