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

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

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


17. Consider the system of equations below.

3x + 2y = 1
2y + z = 2
2x - 2z = -6 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

(1) 1
(2) -1
(3) -4
(4) 4

Answer: (1) 1

Subtract the second equation from the first equation and the y variable will disappear. Now you can solve for x.

3x + 2y = 1
2y + z = 2
2x - 2z = -6

3x - z = -1
2x - 2z = -6

6x - 2z = -2
2x - 2z = -6

4x = 4

x = 1

This is Choice (1).

Checking: 2(1) - 2z = -6; -2z = -8, z = 4

Then, 2y + 4 = 2, 2y = -2, y = -1

Finally, 3(1) + 2(-1) = 1. Check!

18. The point (2,–3) lies on the graph of the equation y = f(x). Which point must lie on the graph of the equation y = f(x - 4) + 1?

(1) (1,1)
(2) (-2,-2)
(3) (3, 7)
(4) (6,-2)

Answer: (4) (6,-2)

The -4 inside the parentheses shifts the graph four units to the right. The +1 outside the parentheses shifts the graph on unit up.

If you translate the point (2,-3) four units to the right and one unit up, you end up at (6,-2).

The correct answer is Choice (4).

19. Which statement best describes the end behavior of the function y = log(x - 3)?

(1) As x → -∞, y → -∞, and as x → ∞, y → ∞.
(2) As x → 3, y → -∞, and as x → ∞, y → ∞.
(3) As x → -∞, y → 0, and as x → ∞, y → ∞.
(4) As x → 3, y → 0, and as x → ∞, y → ∞.

Answer: (2) As x → 3, y → -∞, and as x → ∞, y → ∞.

Graph the function. One the left side, the graph goes to negative infinity as it gets closer to 3. On the right side, both x and y tend toward infinity.

Choice (2) is the correct answer.

20. The black bear population for a certain area of the Adirondacks can be modeled by the equation

B = 5835.943(1.026)^t, where t is measured in years since 2010. Kieran would like to rewrite this model in terms of a 5-year growth rate. Kieran’s model is best represented by

(1) B = 5835.943(1.005147)^t/5
(2) B = 5835.943(1.005147)^5t
(3) B = 5835.943(1.136938)^t/5
(4) B = 5835.943(1.136938)^5t

Answer: (3) B = 5835.943(1.136938)^t/5

If there is a 5-year growth rate, then the exponent will be divided by 5, or t/5. Eliminate Choices (2) and (4).

(1.026)^t is equal to (1.026⁵)^t is equal to (1.136938)^t/5.

This is Choice (3).

21. Which expression or expressions are equal to 0 for all real numbers?

I. (x² + y²)² + (x² + y²)² - 2(x² + y²)² II.(x² + y²)² - (x² + y²)² III. (x² + y²)² - (x² + y²)² - (2xy)²

(1) I, only
(2) III, only
(3) I and II, only
(4) I and III, only

Answer: (4) I and III, only

The first expression is "obviously" equal to 0 (see below), but you need to expand the second and third choices to see if all the terms will cancel out. Again, it is "obvious" that they both can't be true. However, it is possible that neither one is true.

In the first expression, let z = (x² + y²)². So z + z - 2z = 0, which is true. Eliminate Choice (2).

In the second expression, the first term is not the same as the second term, so subtracting them cannot equal 0. Eliminate Choice (3).

Expand the third expressiong.

(x² + y²)² - (x² + y²)² - (2xy)²
(x⁴ + 2x²y² + y⁴) - (x⁴ - 2x²y² + y⁴) - (4x²y²)
x⁴ + 2x²y² + y⁴ - x⁴ + 2x²y² - y⁴ - 4x²y²
x⁴ - x⁴ + 2x²y² + 2x²y² + y⁴ - 4x²y² - y⁴ = 0

Choice (4) is the correct answer.


22. The equation 1/x - 1/5 = x/5 has

(1) rational solutions
(2) irrational solutions
(3) imaginary solutions
(4) no solutions

Answer: (2) irrational solutions

Solve for x.

1/x - 1/5 = x/5

1/x = x/5 + 1/5

1/x = (x + 1)/5

x(x + 1) = 5

x² + x = 5

x² + x - 5 = 0

If you check the discriminant, b² - 4ac, then 1² - (4)(1)(-5) = 21, which is positive but not a perfect square. This means that there are two solutions but they will be irrational.

This is Choice (2).

23. For x ≠ +4y, the expression (x² + 3xy - 28y²) / (16y² - x² is equivalent to
(1) -1 - (7/4) y
(2) (x - 7y) / (4y - x)
(3) (x + 7y) / (x + 4y)
(4) (-x - 7y) / (x + 4y)

Answer: (4) (-x - 7y) / (x + 4y)

Factor the numerator and the denominator and simplify. (x² + 3xy - 28y²) / (16y² - x²
( (x + 7y)(x - 4y) ) / ( (4y - x)(4y + x) )
( (x + 7y)(-1) ) / (4y + x)
( -x - 7y) / (x + 4y)

This is Choice (4).

24. Which equation represents a parabola with a focus of (-2,1) and directrix of y = 5?,

(1) (x + 2)² = -8(y - 3)
(2) (x + 2)² = 5(y - 1)
(3) (x + 2)² = -8(y - 1)
(4) (x + 2)² = 8(y - 3)

Answer: (1) (x + 2)² = -8(y - 3)

The formula for finding a parabola from the focus and directrix is (x - h)² = 4p(y - k), where p is the distance from the vertex to the focus. The vertex is halfway between the focus and directrix, or (-2, 3), which makes p = -2.

Substituting what we know, we get (x - (-2))² = 4(-2)(y - 3), or (x + 2)² = -8(y - 3).

This is Choice (1).

End of Part I.

Questions, comments, and corrections welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in June 2025.

More Regents problems.

June 2025 Algebra 2 Regents

Part I

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


9. The probabilities that a randomly selected teenager uses social media websites F and I are shown below.

P(F) = 0.71
P(I) = 0.52
P(F or I) = 0.77 Given this information, what is P(F and I), the probability that a randomly selected teenager uses both websites?

(1) 0.06
(2) 0.19
(3) 0.46
(4) 0.96

Answer: (3) 0.46

The union (OR) of two probabilities is the equal to the sum of the independent probabilities minus their intersection (AND).

P(F or I) = P(F) + P(I) - P(F and I)
P(F) + P(I) - P(F or I) = P(F and I)
0.71 + 0.52 - 0.77 = P(F and I)
0.46 = P(F and I)

This is Choice (3).

10. Consider f(x) = (x - 2)²(x + 3), and g(x) as strictly defined in the table below.
Which statement or statements must be true, based on the information given?
I. Both f(x) and g(x) have the same x-intercepts.
II. Both f(x) and g(x) have a y-intercept at y = -6

(1) I, only
(2) II, only
(3) I and II
(4) neither I or II

Answer: (1) I, only

The x-intercepts of f(x) are 2 and -3. The x-intercepts of g(x) are -3 and 2. These are the same.

The y-intercept of f(x) = (-2)²(3) = 12. The y-intercept of g(x) is -6. These are not the same.

The correct answer is Choice (1).

11. Josie examines the graphs of f(x) = 3^x - 8 and g(x) = 1 / (x² - 4). The number of solutions to f(x) = g(x) is

(1) 1
(2) 2
(3) 3
(4) 0

Answer: (3) 3

Just graph them. The answer is 3.

Choice (3) is the correct answer.

12. Which binomial is a factor of g³ + 6g² + g - 14?

(1) g - 1
(2) g - 2
(3) g + 1
(4) g + 2

Answer: (4) g + 2

Once again: graph it. Check the zeroes. The roots are x = -2, or x = -2 + √(11).

If g - 1 is a factor, then the polynomial will eqaul zero when g = 1. However, it is -6. Eliminate Choice (1).

If g - 2 is a factor, then the polynomial will eqaul zero when g = 2. However, it is 20. Eliminate Choice (2).

If g + 1 is a factor, then the polynomial will eqaul zero when g = -1. However, it is -10. Eliminate Choice (3).

If g + 2 is a factor, then the polynomial will eqaul zero when g = -2. And it is 0. Choice (4) is the correct answer.

13. Consider the recursively defined sequence below.

a₁ = 8
a_n = 2a_n-1 Which explicit formula represents the same sequence?

(1) a_n = 2ⁿ
(2) a_n = 2(4ⁿ)
(3) a_n = 2⁽ⁿ⁺²⁾
(4) a_n = 8ⁿ

Answer: (3) a_n = 2⁽ⁿ⁺²⁾

If the answer doesn't jump out at you, write out the first few terms of the sequence. Then check each one.

The recursive formula is doubling, starting with 8: 8, 16, 32, 64, ...

Choice (1) is 2, 4, 8, ... Eliminate Choice (1).

Choice (2) is 8, 32, 128, ... Eliminate Choice (2).

Choice (3) is 8, 16, 32, 64. ... Choice (3) is the correct answer.

Choice (4) is 8, 64 ... Eliminate Choice (4).


14. What is the exact value of tan(-5π/6)?

(1) 1 / √(3)
(2) -1 / √(3)
(3) √(3)
(4) -√(3)

Answer: (1) 1 / √(3)

Tangent is positive in Quadrants I and III and this is Quadrant III. Eliminate Choices (2) and (4).

The coordinates for that point on the unit circle are (-√(3)/2, -1/2).

Tangent is sin/cos = (-1/2) / (-√(3)/2) = 1 / √(3)

This is Choice (1).

15. Given m ≠ 0 and (17^1/m)ⁿ = 17², what is n in terms of m?
(1) 2m
(2) 2/m
(3) m/2
(4) 2^m

Answer: (1) 2m

Multiplying the exponents give you 17^n/m = 17². This means n/m = 2.

Therefore, n = 2m.

This is Choice (1).

16. In order to qualify for a college tennis scholarship, Joe needs to win 90% of the matches he plays during his senior year of high school. If he has won 8 of the 10 matches that he has played, which equation can be used to determine how many more consecutive matches, x, Joe must win in order for his winning percentage to equal 90%?

(1) (8 + x) / x = 0.90
(2) 8 / (10 + x) = 0.90
(3) 8/10 + x = 0.90
(4) (8 + x) / (10 + x) = 0.90

Answer: (4) (8 + x) / (10 + x) = 0.90

The number of games won divided by the number of games played must equal 0.90.

Currently, Joe is at 8/10 = .80. If he keeps winning, his average will be 9/11, 10/12, 11/13, etc.

Both the numerator and the denominator will increase.

This means that x games will be added to both the 8 and the 10, so Choice (4) is the correct answer.

More to come. Comments and questions welcome.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

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

This exam was adminstered in June 2025.

More Regents problems.

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.

I also write Fiction! The NEW COLLECTION IS AVIALABLE! A Bucket Full of Moonlight, written by Christopher J. Burke, contains 30+ pieces of short stories and flash fiction. It's available from eSpec Books! Order the softcover or ebook at Amazon. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants.
My other books include my Burke's Lore Briefs series and In A Flash 2020.
If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!