AIME Algebra 1 — Questions and Answers
Question 1: Find the sum of all real values of x satisfying x^4 - 5x^2 + 4 = 0.
- 0 (Correct answer)
- 2
- 4
- 5
Correct answer: 0
The equation factors as (x^2-1)(x^2-4)=0 giving x=±1, ±2, whose sum is 0.
Question 2: If a + b = 5 and ab = 3, what is a^3 + b^3?
- 80 (Correct answer)
- 95
- 110
- 65
Correct answer: 80
Using the identity a^3+b^3=(a+b)^3-3ab(a+b)=125-45=80.
Question 3: The system 2x + 3y = 7 and 4x + ky = 14 has infinitely many solutions when k equals:
- 6 (Correct answer)
- 3
- 7
- 4
Correct 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.
Question 4: If f(x) = x^2 + 2x + 1 and g(x) = x - 1, find f(g(3)).
- 4 (Correct answer)
- 9
- 16
- 1
Correct 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.
Question 5: What is the product of all solutions to (x-2)^2 = 9?
- -5 (Correct answer)
- 5
- 13
- -13
Correct answer: -5
Solving gives x=5 or x=-1; by Vieta's the product of roots of x^2-4x-5=0 is -5.
Question 6: If log_2(x) + log_2(x-2) = 3, find x.
- 4 (Correct answer)
- 2
- 8
- 6
Correct 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).
AIME Algebra 1
Find the sum of all real values of x satisfying x^4 - 5x^2 + 4 = 0.