SAT Math: Rewrite x² - 10x + 14 in Vertex Form (x - h)² + k, Find h + k
(x^2 - 10x + 25) - 25 + 14 = (x - 5)^2 - 11. h = 5, k = -11. h + k = 5 + (-11) = -6.
Exact Question Text
The expression x^2 - 10x + 14 is equivalent to (x - h)^2 + k. What is the value of h + k?
1. Step-by-Step Formal Solution
- Complete square: (x - 10/2)^2 = (x - 5)^2 = x^2 - 10x + 25.
- (x - 5)^2 - 25 + 14 = (x - 5)^2 - 11.
- h = 5, k = -11 => h + k = -6.
2. Fast Desmos / Calculator Shortcut
Graph y = x^2 - 10x + 14. Vertex is (5, -11). h + k = 5 - 11 = -6.
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.
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(x^2 - 10x + 25) - 25 + 14 = (x - 5)^2 - 11. h = 5, k = -11. h + k = 5 + (-11) = -6.
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