AIME Algebra

Question 1

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

Accepted Answer

0

Answer

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?

Accepted Answer

80

Answer

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:

Accepted Answer

6

Answer

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)).

Accepted Answer

4

Answer

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?

Accepted Answer

-5

Answer

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

AIME - American Invitational Mathematics Examination Practice Test