SAT Math Practice Question #676 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #676

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The equation \[ \frac{2x}{x-3} + \frac{1}{x+a} = \frac{4x^2 - 5x + 3}{x^2 + (a-3)x - 3a} \] has exactly one valid real solution for \( x \). If \( a \) is a positive integer, what is the value 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 an elite 800-level algebra question testing rational equations and the concept of extraneous solutions. Step 1: Notice the denominator on the right side. If you factor \( x^2 + (a-3)x - 3a \), you get \( (x-3)(x+a) \). This is exact...

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

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