SAT Math: If 9x² - 64 = (ax + b)(ax - b), What Is the Value of a + b?

Exact Question Text

If 9x^2 - 64 = (ax + b)(ax - b) for all values of x, where a and b are positive constants, what is the value of a + b?

1. Step-by-Step Formal Solution

1. Difference of squares: (3x)^2 - (8)^2 = (3x + 8)(3x - 8).
2. a = 3, b = 8 => a + b = 11.

2. Fast Desmos / Calculator Shortcut

Evaluate sqrt(9) + sqrt(64) = 3 + 8 = 11.

3. Distractor Trap Analysis

• Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
• Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
• Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
• Option Correct: Accurately satisfies all algebraic and logical constraints.