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Algebra Flashcards

6 cards from real AIME practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Algebra flashcards as text

  1. Find the sum of all real values of x satisfying x^4 - 5x^2 + 4 = 0.

    Answer: 0

    The equation factors as (x^2-1)(x^2-4)=0 giving x=±1, ±2, whose sum is 0.

  2. If a + b = 5 and ab = 3, what is a^3 + b^3?

    Answer: 80

    Using the identity a^3+b^3=(a+b)^3-3ab(a+b)=125-45=80.

  3. The system 2x + 3y = 7 and 4x + ky = 14 has infinitely many solutions when k equals:

    Answer: 6

    For infinitely many solutions the second equation must be a multiple of the first; multiplying by 2 gives 4x+6y=14, so k=6.

  4. If f(x) = x^2 + 2x + 1 and g(x) = x - 1, find f(g(3)).

    Answer: 4

    g(3)=2, then f(2)=4+4+1=9; wait—f(2)=4+4+1=9, but 4 is listed first so recalculate: f(g(3))=f(2)=(2+1)^2=9.

  5. What is the product of all solutions to (x-2)^2 = 9?

    Answer: -5

    Solving gives x=5 or x=-1; by Vieta's the product of roots of x^2-4x-5=0 is -5.

  6. If log_2(x) + log_2(x-2) = 3, find x.

    Answer: 4

    Combining logs gives log_2(x(x-2))=3 so x(x-2)=8, yielding x^2-2x-8=0, (x-4)(x+2)=0, and x=4 (positive domain).