GRE Algebraic Equations and Functions 3

Question 1

Solve the system: x + y = 10 and x - y = 4. What is x?

Accepted Answer

7

Answer

Adding the equations gives 2x = 14, so x = 7.

Question 2

Factor completely: x² - 9.

Accepted Answer

(x - 3)(x + 3)

Answer

This is a difference of squares: x² - 9 = (x - 3)(x + 3).

Question 3

If f(x) = x² - 4x + 4, what are the roots?

Accepted Answer

x = 2 (double)

Answer

x² - 4x + 4 = (x - 2)², so x = 2 is a repeated root.

Question 4

Simplify: (2x³)(3x²).

Accepted Answer

6x⁵

Answer

Multiply coefficients (2·3 = 6) and add exponents (3 + 2 = 5) to get 6x⁵.

Question 5

If 3^x = 81, what is x?

Accepted Answer

4

Answer

Since 81 = 3⁴, x = 4.

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