If a² + b² + c² = 2(a + c - 1), then find the value of a³ + b³ + c³.

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Accepted Answer

Given: a² + b² + c² = 2(a + c - 1)

Step 1: Rewrite the equation:

a² + b² + c² = 2a + 2c - 2

Step 2: Rearrange terms:

a² - 2a + b² + c² - 2c = -2

Step 3: Complete the square for a and c:

(a² - 2a + 1) + b² + (c² - 2c + 1) = -2 + 1 + 1

(a - 1)² + b² + (c - 1)² = 0

Step 4: Since squares of real numbers are ≥ 0 and their sum is 0, each must be zero:

(a - 1)² = 0 ⇒ a = 1
 b² = 0 ⇒ b = 0
 (c - 1)² = 0 ⇒ c = 1

Step 5: Calculate a³ + b³ + c³:

1³ + 0³ + 1³ = 1 + 0 + 1 = 2

Re-check the calculation:

Given the sum of squares equals 2(a + c - 1), after completing the square, we found (a-1)² + b² + (c-1)² = 0.

This means a=1, b=0, c=1.

Then a³ + b³ + c³ = 1 + 0 + 1 = 2.

Therefore, the correct answer is 2.

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