SOLUTION: hello sir/madam i am shreya from new delhi i want to ask if a,b,c are all non zero and a+b+c=0 prove that a^square/bc+b^square/ca+c^square/ab=3

Question 622302: hello sir/madam i am shreya from new delhi i want to ask if a,b,c are all non zero and a+b+c=0 prove that a^square/bc+b^square/ca+c^square/ab=3
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
There may be a simpler way for this but here goes...

If

then

To cube the left side I am going to use the pattern with the "x" being "(a+b)" and the "y" being "c":

For the I will use this pattern again:

For the I will use the pattern:

Multiplying we get:

Next I'll isolate the cubed terms on the left side by subtracting all the non-cubed terms from both sides:

Factoring out -3 on the right side:

Dividing both sides by abc. (We're told that a b and c are all non-zero so it is OK to divide by this.)

Splitting up the fractions:

Simplifying the fractions:

(Note that the left side is the expression we hope is equal to 3.) Adding the like fractions on the right side:

From a + b + c = 0:

Subtracting c from each side we get: a + b = -c
Subtracting b from each side we get: a + c = -b
Subtracting a from each side we get: b + c = -a

Substituting these expressions in for the numerators on the right side we get:

Simplifying the fractions further:

Simplifying:

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