(x, why?): June 2026

Monday, June 01, 2026

Algebra 2 Problems of the Day (Algebra 2, June 2025 Part I)

This exam was adminstered in June 2025.

June 2025 Algebra 2 Regents

Part I

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

1. Which expression is equivalent to 2c³√(c) ?

(1) 2c^4/3
(2) 2c^3/4
(3) (2c)^4/3
(4) (2c)^3/4

Answer: (1) 2c^4/3

When there's a fraction as an exponent, the numerator is the power the base is raised to while the denominator is the nth root of the base.

2c³√(c) = 2c (c)^1/3

When you multiply variables, keep the base and add the exponents. If you add 1 and 1/3, the sum is 4/3. Therefore,

2c³√(c) = 2c (c)^1/3 = 2c^4/3

The correct answer is Choice (1).

2. Which investigation technique is most often used to determine the cause and effect of a medication?

(1) observational study
(2) survey
(3) controlled experiment
(4) census

Answer: (3) controlled experiment

To determine cause and effect, you need to conduct a controlled experiement. A survey, census or observational study are insufficient.

The correct answer is Choice (3).

3. What is the solution to 5(2)^19x = 50?

(1) x = (log(50)) / 19
(2) x = (log₂(10)) / 19
(3) x = (log₂(45)) / 19
(4) x = 5 / 19

Answer: (2) x = (log₂(10)) / 19

Use inverse operations.

5(2)^19x = 50
(2)^19x = 10
log₂ 10 = 19x
x = log₂ 10 / 19

The correct answer is Choice (2).

4. The function P(t) = 256,485(0.965)^t models the decreasing population of a city from 1999 to 2014, where t is the time in years since 1999.

Which statement is not true?

(1) The function estimated the population was 256,485 in 1999.
(2) The decay rate was 0.35%.
(3) The decay factor is 0.965.
(4) The population declined over 15 years.

Answer: (2) The decay rate was 0.35%.

I will say that this was a confusing question, and for a moment, I wondered if there was a typo. However, I would've remembered if a recent question had been invalidated. My trouble comes from them using the term "decay factor", which I have to say, personally, I don't see that term used. Note that I don't normally teach Algebra II, but when talking about equations of this type, we usually talk about the "decay rate" and not the "decay factor".

In any case ...

In Choice (1), the starting population is 256,485 in 1999. This is true, so eliminate Choice (1).

In Choice (2), the decay rate is NOT 0.35%. If you subtract 1.000 - 0.965, you get 0.035, which is 3.5%. This is the correct answer.

In Choice (3), the decay factor is 0.965. This is true. This is different from the decay rate. This is the factor that you are multiply the starting value by. Eliminate Choice (3).

In Choice (4), since the decay factor is less than 1.00, which is why it's decay and not growth, the population will continually shrink. The question says it's decreasing from 1999 to 2014, so it's decreasing for 15 years. This is correct, so eliminate Choice (4).

5. Four different surveys gathered data about the purchasing behaviors of pet owners. Pet owners from the same population were randomly selected. While collecting data, Chris surveyed 942 pet owners, John surveyed 410, Brooke surveyed 800, and Shane surveyed 100. Whose survey will likely have the smallest margin of error?

(1) Brooke
(2) Chris
(3) John
(4) Shane

Answer: (2) Chris

The more pet owners interviewed, the smaller the margin of error that can be expected. More is better.

Chris interviewed the most. This is Choice (2).

6. Given i is the imaginary unit and a = i³, b = i², and c = i, which expression is equivalent to 2ax² + 3bx - cx?

(1) -2ix² - 3x + ix
(2) -2ix² - 3x
(3) -2ix² - 3x - ix
(4) -8ix² - 3x - ix

Answer: (3) -2ix² - 3x - ix

Replace a with (-i). Replace b with (-1). Replace c with (i). Simplify the expression.

2ax² + 3bx - cx = 2(-i)x² + 3(-1)x - (i)x = -2ix² - 3x - ix.

This is Choice (3).

7. Which sequence has a common ratio of 1/2?

(1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...
(2) 1/32 a, 1/16 a, 1/8 a, 1/4 a, ...
(3) 20a, 39/2 a, 19a, 37/2 a, ...
(4) 22a, 22.5a, 23a, 23.5a, ...

Answer: (1) -1/4 a, -1/8 a, -1/16 a, -1/32 a ...

Multiply the first term by 1/2. Did you get the second term? If not, eliminate that choice. If you did, check the next number.

In Choice (1), if you multiply (-1/4a)(1/2) = -1/8 a, etc. This is the Correct answer.

In Choice (2), if you multiply (1/32 a)(1/2) = 1/64 a. The common ratio in the second choice is 2, not 1/2. Eliminate Choice (2).

In Choice (3), half of 20a is 10a, not 39/2 a. This is an arithmetic sequence with a common difference of -1/2 a.

In Choice (4), half of 22a is 11a, not 22.5a. This is an arithmetic sequence with a common difference of 1/2 a, or .5a.

8. The result of dividing 2x³ + 6x² + 7x + 2 by x + 1 is

(1) 2² + 4x + 3 - 1 / (x + 1)
(2) 2² + 4x + 3 + 1 / (x + 1)
(3) 2² + 8x - 15 + 17 / (x + 1)
(4) 2² + 8x + 15 - 13 / (x + 1)
*******************************

Answer: (1) 2² + 4x + 3 - 1 / (x + 1)

If 2x³ + 6x² + 7x + 2 were divisible by (x + 1), then f(-1) would be equal to 0. However, f(-1) = -1, which is a remainder.

Only one choice has -1 / (x + 1) in it, so that is the answer.

Doing it the long way:

More to come. Comments and questions welcome.

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