Algebra & Functions Flashcards
7 cards from real AMC 12 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 Algebra & Functions flashcards as text
If a + b = 5 and ab = 3, what is a^2 + b^2?
Answer: 19
a^2 + b^2 = (a + b)^2 - 2ab = 25 - 6 = 19.
What is the minimum value of f(x) = |x - 2| + |x + 4|?
Answer: 6
The sum of distances from x to 2 and to -4 is minimized at 6 for any x between them.
How many real solutions does x^4 - 5x^2 + 4 = 0 have?
Answer: 4
It factors as (x^2 - 1)(x^2 - 4) = 0, giving x = ±1, ±2.
If f(x) + 2f(1 - x) = x for all real x, what is f(0)?
Answer: 2/3
Plugging x = 0 and x = 1 gives f(0) + 2f(1) = 0 and f(1) + 2f(0) = 1, so f(0) = 2/3.
What is the sum of the infinite series 1 + 1/3 + 1/9 + 1/27 + ...?
Answer: 3/2
An infinite geometric series sums to a/(1 - r) = 1/(2/3) = 3/2.
If r and s are the roots of x^2 - 5x + 2 = 0, what is 1/r + 1/s?
Answer: 5/2
1/r + 1/s = (r + s)/(rs) = 5/2.
Solve 4^x - 3·2^x - 4 = 0 for real x.
Answer: 2
With u = 2^x, u^2 - 3u - 4 = 0 gives u = 4 (u = -1 is impossible), so x = 2.
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