If a, b, c are real numbers and a² + b² + c² = 2(a - b - c) - 3, then the value of a + b + c is

Given:

a2 + b2 + c2 = 2(a - b - c) - 3

Concept used:

1. A² + B² + C² = 0 then A = B = C = 0

2. (a - b)² = a² - 2ab + b²

3. (a + b)² = a² + 2ab + b²

Calculation:

a2 + b2 + c2 = 2(a - b - c) - 3

⇒ (a² - 2a + 1) + (b² + 2b + 1) + (c² + 2c + 1) = 0

⇒ (a - 1)² + (b + 1)² + (c + 1)² = 0

According to the concept,

(a - 1)2 = 0

⇒ a = 1

(b + 1)2 = 0

⇒ b = -1

(c + 1)2 = 0

⇒ c = -1

Now, a + b + c

⇒ 1 - 1 - 1 = -1

∴ The required value of a + b + c is -1.

Alternate MethodGiven:

a, b, c are real numbers and a² + b² + c² = 2(a - b - c) - 3

Calculations:

Rearrange the equation:

a² + b² + c² = 2a - 2b - 2c - 3

⇒ a² - 2a + b² + 2b + c² + 2c + 3 = 0

Group terms to form perfect squares:

⇒ (a² - 2a + 1) + (b² + 2b + 1) + (c² + 2c + 1) = 0

⇒ (a - 1)² + (b + 1)² + (c + 1)² = 0

Since a, b, c are real numbers, each squared term must be zero for their sum to be zero:

a - 1 = 0 ⇒ a = 1

b + 1 = 0 ⇒ b = -1

c + 1 = 0 ⇒ c = -1

Find a + b + c:

a + b + c = 1 + (-1) + (-1)

⇒ a + b + c = 1 - 1 - 1

⇒ a + b + c = -1

∴ The value of a + b + c is -1.