(x, why?): Algebra Problems of the Day (Algebra Regents, August 2024 Part I)

This exam was adminstered in August 2024.

August 2024 Algebra Regents

Part I

Each correct answer will receive 2 credits. No partial credit.

1.What is the correct factorization of x² + 4x - 12?

(1) (x + 3)(x - 4)
(2) (x - 3)x + 4)
(3) (x + 2)(x - 6)
(4) (x - 2)(x + 6)

Answer: (4) (x - 2)(x + 6)

What two factors of -12 have a sum of +4? One factor must be negative and the other must be positive, and the "larger" number must be positive.

Choice (1) is x² - x - 12.

Choice (2) is x² + x - 12.

Choice (3) is x² - 4x - 12.

Choice (2) is x² + 4x - 12. This is the correct answer.

2. Which situation can be modeled by a linear function?

(1) A printer can print one page every three seconds.
(2) A bank account earns 0.5% interest each year, compounded annually.
(3) The number of cells in an organism doubles every four days.
(4) The attendance at a professional sports team’s games decreases by 1.5% each year.

Answer: (1) A printer can print one page every three seconds.

The linear function never changes. It has a constant rate. One page every three seconds is a constant rate of change for the total number of pages printed. The correct answer is Choice (1).

Choice (2) is not linear because the interest is compounded, meaning interest is earned on the interest.

Choice (3) is not linear because the amount doubles every day, which is not a constant rate of range.

Choice (4) is not linear because the decrease is compounded from year to year, losing 1.5% of the attendance of the year before.

3. Which expression is equivalent to 3(x² - 2x + 3) - (4x² + 3x - 1)?

(1) -x² + x + 2
(2) -x² - 8x + 7
(3) -x² - 3x + 8
(4) -x² - 9x + 10

Answer: (4) -x² - 9x + 10

Distribute the 3 and the minus sign and then combine like terms.

3(x² - 2x + 3) - (4x² + 3x - 1)
3x² - 6x + 9 - 4x² - 3x + 1
-x² - 9x + 10

This is Choice (4).

4. At Adelynn’s first birthday party, each guest brought $1 in coins for her piggy bank. Guests brought nickels, dimes, and quarters for a total of $28. There were twice as many dimes as nickels and 12 more quarters than nickels. Which equation could be used to determine the number of nickels, x, that her guests brought to her party?

(1) .05x + .10x + .25x = 28
(2) .05x + .10(2x) + .25(x + 12) = 28
(3) .05(2x) + .10x + .25(x + 12) = 28
(4) .05(x + 12) + .10(2x) + .25x = 28

Answer: (2) .05x + .10(2x) + .25(x + 12) = 28

If x represents the number of nickels, then the value of the nickels is .05x. Eliminate Choices (3) and (4).

Choices (1) states that there are the same number of nickels, dimes, and quarters, which is not true. Eliminate Choice (1).

Choice (2) is the correct answer because there are 2 times as many dimes as nickels and 12 more quarters.

5. A student creates a fourth-degree trinomial with a leading coefficient of 2 and a constant value of 5. The trinomial could be

(1) 2x⁴ + 3x² + 5
(2) 2x⁴ + 5x + 3
(3) 4x² - 3x + 5
(4) 4x³ - 5x² + 3

Answer: (1) 2x⁴ + 3x² + 5

Fourth-degree means that the polynomial has x⁴ as the term with the largest exponent. Leading coefficient of 2 means that it will have 2x⁴. The constant has no variable and has to equal 5.

The only possibility is Choice (1).

Choice (2) has a constant of 3, not 5.

Choices (3) and (4) do not have a leading coefficient of 2, nor are they fourth-degree.

6. When solving the equation 4x² - 16 = 0, Laura wrote 4x² = 16 as her first step. Which property justifies Laura’s first step?

(1) distributive property of multiplication over addition
(2) multiplication property of equality
(3) commutative property of addition
(4) addition property of equality

Answer: (4) addition property of equality

Whether or not this is how you would solve this equation, the question is about Laura solving the problem. The first thing she did was add 16 to both sides of the equation. She can do this because of the Addition Property of Equality.

The correct answer is Choice (4).

7. Which expression results in an irrational number?

Answer: (4) 1/3 + √(3)

The sum of a rational and an irrational number is always irrational.

Choice (1) has a value of 3, which is rational.

Choice (2) has a value of 45, which is rational.

Choice (3) has a value of -5/12, which is rational.

8. Which equation has the same solutions as x² + 6x - 18 = 0?

(1) (x + 3)² = 24
(2) (x + 3)² = 27
(3) (x + 6)² = 24
(4) (x + 6)² = 27

Answer: (2) (x + 3)² = 27

Complete the Square:

x² + 6x - 18 = 0
x² + 6x = 18
x² + 6x + 9 = 27
(x + 3)² = 27

Half of 6 is 3 and 3 squared is 9. Also, x² + 6x + 9 is a perfect square, namely (x + 3)².

The correct choice is Choice (2).

More to come. Comments and questions welcome.

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