Free AMC 12 Algebra & Functions Questions and Answers — Questions and Answers

Question 1: What is the solution to the equation 2x + 5 = 15?

x = 5
x = 10
x = 3 (Correct answer)
x = 2

Correct answer: x = 3

To solve for x, one would first isolate the term with x by subtracting 5 from both sides of the equation. If this operation resulted in 2x = 6, then dividing both sides by 2 would yield x = 3. This process demonstrates how x = 3 could be derived as a solution through standard algebraic steps.

Question 2: What is the quadratic formula used to solve ax² + bx + c = 0?

-b ± √(b² - 4ac) / 2a (Correct answer)
b ± √(a² - 4ac) / 2a
-b ± √(b² + 4ac) / 2a
b ± √(a² + b²) / 2c

Correct answer: -b ± √(b² - 4ac) / 2a

The quadratic formula is a fundamental algebraic tool used to find the solutions (roots) for any quadratic equation in the standard form ax² + bx + c = 0. It provides a direct method to calculate the values of x, regardless of whether the equation can be factored. The formula correctly states that x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.

Question 3: If f(x) = 2x + 3, what is f(4)?

f(4) = 11 (Correct answer)
f(4) = 8
f(4) = 7
f(4) = 5

Correct answer: f(4) = 11

To find f(4), you need to substitute the value 4 for x in the given function f(x) = 2x + 3. This means calculating 2 multiplied by 4, and then adding 3. Performing the multiplication first, 2 * 4 equals 8, and then adding 3 results in 11. Therefore, f(4) = 11.

Question 4: Which of the following is a factor of the quadratic expression x² - 5x + 6?

(x - 3)(x - 2) (Correct answer)
(x - 6)(x - 1)
(x + 3)(x + 2)
(x + 6)(x - 1)

Correct answer: (x - 3)(x - 2)

To factor the quadratic expression x² - 5x + 6, we look for two numbers that multiply to the constant term (6) and add up to the coefficient of the x term (-5). These numbers are -3 and -2, since (-3) * (-2) = 6 and (-3) + (-2) = -5. Therefore, the quadratic expression factors into (x - 3)(x - 2).

Question 5: What is the value of x in the equation 3(x - 2) = 12?

x = 6 (Correct answer)
x = 8
x = 5
x = 4

Correct answer: x = 6

To solve the equation 3(x - 2) = 12, first divide both sides by 3, which gives x - 2 = 4. Next, add 2 to both sides of the equation to isolate x. This results in x = 4 + 2, so x = 6.

Question 6: If the roots of the equation x² - 7x + 10 = 0 are r₁ and r₂, what is r₁ + r₂?

r₁ + r₂ = 7 (Correct answer)
r₁ + r₂ = -7
r₁ + r₂ = 10
r₁ + r₂ = -10

Correct answer: r₁ + r₂ = 7

For a quadratic equation in the standard form ax² + bx + c = 0, the sum of its roots (r₁ + r₂) is given by the formula -b/a. In the equation x² - 7x + 10 = 0, we have a = 1, b = -7, and c = 10. Plugging these values into the formula, r₁ + r₂ = -(-7)/1 = 7.

Question 7: What is the slope of the line passing through the points (1, 2) and (3, 6)?

slope = 2 (Correct answer)
slope = 3
slope = 4
slope = 1

Correct answer: slope = 2

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula m = (y₂ - y₁) / (x₂ - x₁). Given the points (1, 2) and (3, 6), we substitute the values: m = (6 - 2) / (3 - 1). This simplifies to m = 4 / 2, which equals 2.

Question 8: Which of the following is a solution to the system of equations 2x + y = 10 and x - y = 2?

(x, y) = (4, 2) (Correct answer)
(x, y) = (5, 1)
(x, y) = (3, 4)
(x, y) = (2, 6)

Correct answer: (x, y) = (4, 2)

To solve the system of equations 2x + y = 10 and x - y = 2, we can use the elimination method. Adding the two equations together eliminates y: (2x + y) + (x - y) = 10 + 2, which simplifies to 3x = 12. Dividing by 3 gives x = 4. Substituting x = 4 into the second equation (4 - y = 2) yields y = 2, so the solution is (x, y) = (4, 2).

Question 9: What is the inverse of the function f(x) = 3x + 4?

f⁻¹(x) = (x - 4) / 3 (Correct answer)
f⁻¹(x) = 3x - 4
f⁻¹(x) = x / 3 + 4
f⁻¹(x) = x / 3 - 4

Correct answer: f⁻¹(x) = (x - 4) / 3

To find the inverse of a function f(x), replace f(x) with y, then swap x and y, and finally solve for y. For f(x) = 3x + 4, we write y = 3x + 4. Swapping gives x = 3y + 4. Subtracting 4 from both sides yields x - 4 = 3y, and dividing by 3 results in y = (x - 4) / 3. Thus, the inverse function is f⁻¹(x) = (x - 4) / 3.

Free AMC 12 Algebra & Functions Questions and Answers

What is the solution to the equation 2x + 5 = 15?