SAT Math Practice Question #29 (Hard (650)) | Test Citadel
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SAT Math Difficulty: Hard (650)

Digital SAT Math Practice Question #29

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

The quadratic function f is defined by f(x) = x^2 - 6x + c, where c is a constant. If the graph of y = f(x) in the xy-plane has exactly one distinct real x-intercept, what is the value of c?
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Socratic AI Engine Step-by-Step Derivation
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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

A quadratic function has exactly one real zero (x-intercept) when its discriminant is equal to zero. The discriminant is b^2 - 4ac. Here, a = 1, b = -6, and c = c. Setting the discriminant to zero: (-6)^2 - 4(1)(c) = 0. This simplifies to 3...

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

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