SAT Math Practice Question #607 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #607

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

The quadratic function \( p(x) \) has its vertex at the point \( (h, k) \) and intersects the \( x \)-axis at exactly two points: \( (2, 0) \) and \( (12, 0) \). The function \( p(x) \) can be written in the form \( p(x) = a(x - h)^2 + k \), where \( a \) is a non-zero constant. A new function, \( q(x) = p(x) + c \), is defined such that it has exactly one \( x \)-intercept. Which of the following expressions represents the value of the constant \( c \) in terms of \( a \)?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Identify the quadratic form: consider converting to vertex form \( y = a(x-h)^2 + k \) or using \( x = - rac{b}{2a} \) to find the line of symmetry.

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Plot the left-hand side and right-hand side as separate equations (e.g. \( y_1 = f(x) \) and \( y_2 = g(x) \)). Click the intersection points directly to read exact coordinates.

Masterclass Solution & Distractor Trap Analysis

This is a beautiful conceptual question about quadratics, transformations, and their roots. Let's walk through the logic step-by-step. Step 1: Understand the original function \( p(x) \). We are given that \( p(x) \) has \( x \)-intercepts ...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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