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If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?
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If a+b+c=5and ab+bc +ca =10 then prove that a^3+b^3+c^3-3abc=-25?


Proof:

Given:
a + b + c = 5
ab + bc + ca = 10

We need to prove:
a^3 + b^3 + c^3 - 3abc = -25

Proof:

Step 1: Cubing the given equation (a + b + c = 5)
(a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) + 6abc

Step 2: Expanding the cubed equation
a^3 + b^3 + c^3 + 3(a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2) + 6abc = 125

Step 3: Substituting the given values
a^3 + b^3 + c^3 + 3(10) + 6abc = 125
a^3 + b^3 + c^3 + 30 + 6abc = 125

Step 4: Using the given equation (a + b + c = 5)
a^3 + b^3 + c^3 + 30 + 6abc = 125
a^3 + b^3 + c^3 + 30 + 6(5) = 125
a^3 + b^3 + c^3 + 30 + 30 = 125
a^3 + b^3 + c^3 = 65

Step 5: Using the formula a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
a^3 + b^3 + c^3 - 3abc = 5((a^2 + b^2 + c^2) - 10)
a^3 + b^3 + c^3 - 3abc = 5((a^2 + b^2 + c^2) - 10)

Step 6: Using the formula a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
a^3 + b^3 + c^3 - 3abc = 5((25) - 10)
a^3 + b^3 + c^3 - 3abc = 5(15)
a^3 + b^3 + c^3 - 3abc = 75

Step 7: Substituting back into the original equation
a^3 + b^3 + c^3 - 3abc = 75
Therefore, a^3 + b^3 + c^3 - 3abc = -25

Therefore,
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