Sharpen your understanding of key algebra and function concepts commonly tested in the GRE Quantitative section. This quiz focuses on equations, polynomials, and function properties to help you assess your skills in algebraic reasoning.
If 3x - 5 = 16, what is the value of x?
Explanation: To solve 3x - 5 = 16, add 5 to both sides to get 3x = 21, then divide by 3 to get x = 7. The option 3.67 results from incorrectly dividing 11 by 3. Option 21 is what you get before dividing by 3. Option 5.5 comes from substituting the wrong values for x. The correct answer is 7 because it satisfies the original equation.
What are the roots of the quadratic equation x^2 - 5x + 6 = 0?
Explanation: Factoring x^2 - 5x + 6 gives (x - 2)(x - 3) = 0, so the roots are x = 2 and x = 3. Option x = -2 and x = -3 comes from changing the signs incorrectly. The options x = 1 and x = 6 and x = -1 and x = -6 do not multiply to 6 nor produce the correct middle term. Thus, x = 2 and x = 3 is correct.
If f(x) = 2x^2 - 3x + 4, what is the value of f(-2)?
Explanation: Substitute x = -2 into the function: f(-2) = 2(-2)^2 - 3(-2) + 4 = 2*4 + 6 + 4 = 8 + 6 + 4 = 18. Option 12 results from incorrectly calculating the squared term. Option 10 is from missing the -3(-2) part. Option 0 is from substituting incorrectly or using wrong signs. The clear answer is 18.
For which value of x does the equation |x - 5| = 3 hold true?
Explanation: The equation |x - 5| = 3 is true when x - 5 = 3 or x - 5 = -3, giving x = 8 or x = 2. Options like x = 3 or x = -3 and x = 8 or x = -8 ignore the translation by 5. The pair x = 3 or x = 5 only matches one condition. Therefore, x = 2 or x = 8 is correct.
If g(x) = x + 4 and h(x) = 2x, what is the value of h(g(3))?
Explanation: First, find g(3): g(3) = 3 + 4 = 7. Next, substitute this into h: h(7) = 2*7 = 14. Option 6 comes from using h(3) instead of h(g(3)), while option 16 is from 2*8 (using x + 5 instead of x + 4). Option 10 uses the wrong output for g(3). Therefore, the correct answer is 14.