Question

If \(a^2 + c^2 + 17 = 2(a - 2b^2 - 8b)\), then the value of \((a + b + c) \left( [(a - b)^2 + (b - c)^2 + (c - a)^2] \right)\) is:

• A. -10
• B. -1
• C. 9
• D. 10

Hint:

When faced with complex algebraic expressions, first simplify the given equation, and then proceed step-by-step to solve for the desired expression.

Answer 1

The Correct Option is D

Given the equation \( a^2 + c^2 + 17 = 2(a - 2b^2 - 8b) \), we first simplify this equation to find a relationship between \(a\), \(b\), and \(c\). We expand both sides: \[ a^2 + c^2 + 17 = 2a - 4b^2 - 16b \] Rearranging the terms: \[ a^2 + c^2 - 2a + 4b^2 + 16b + 17 = 0 \] Now, we substitute values of \(a\), \(b\), and \(c\) that satisfy this equation and calculate the value of: \[ (a + b + c) \left( (a - b)^2 + (b - c)^2 + (c - a)^2 \right) \] Upon solving, the final result for the given expression is 10. Thus, the correct answer is 10.