Algebra Problems of the Day (Algebra Regents, January 2026 Part I)

This exam was adminstered in January 2026.

More Regents problems.

January 2026 Algebra Regents

Part I

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


1. A parabola is graphed on the set of axes below.
What are the equation of the axis of symmetry and the coordinates of the vertex of this parabola?

(1) x = 3 and (3, -4)
(2) y = 3 and (3, -4)
(3) x = -4 and (-4, 3)
(4) y = -4 and (-4, 3)

Answer: (1) x = 3 and (3, -4)

The axis of symmentry is a vertical line that goes through the vertex of the parabola with an equation x = a, where a is the x-coordinate of the vertex.

The vertex is (3, -4) and the axis of symmetry is x = 3.

The correct answer is Choice (1).

2. The product of &sqrt;(25) and &sqrt;(2) will result in

(1) an irrational number
(2) a rational number
(3) a natural number
(4) an integer

Answer: (1) an irrational number

The product of &sqrt;(25) and &sqrt;(2) is &sqrt;(50), which is irrational because 50 is not a perfect square.

The product MUST be either Rational or Irrational, so Choices (3) and (4) could be eliminated because all natural numbers are also integers and all integers are rational numbers.

The correct answer is Choice (1).

3. When f(x) = |4x + 2| and g(x) = 3x + 5 are graphed on the same set of axes, for which value of x is f(x) = g(x)?

(1) 1
(2) 2
(3) 3
(4) 14

Answer: (3) 3

You can plug each choice into each function using the calculator, or just graph them. Or you can solve it.

4x + 2 = 3x + 5
x = 3

4x + 2 = -1(3x + 5)
4x + 2 = -3x - 5
7x = -7
x = -1

The two functions intersect at x = -1 and x = 3.

The correct Choice is Choice (3).

4. The expression x² - 26x - 120 is equivalent to

(1) (x + 4)(x - 30)
(2) (x - 4)(x + 30)
(3) (x - 20)(x + 6)
(4) (x + 20)(x - 6)

Answer: (1) (x + 4)(x - 30)

The factors of the constant term, -120, have to add up to coefficient of the middle term, which is -26. This means that there will be one positive and one negative term and that the negative term must be bigger.

Choice (1) is the correct answer.

Choice (2) gives a middle term of + 26x. Choice (3) gives a middle term of -14x. Choice (4) gives a middle term of -14x.

5. The expression 3 - 2&sqrt;(5) + 6&sqrt;(5) is equivalent to

(1) 7&sqrt;(5)
(2) 7&sqrt;(10)
(3) 3 + 4&sqrt;(5)
(4) 3 + 4&sqrt;(10)

Answer: (3) 3 + 4&sqrt;(5)

You can only combine like terms. Imagine this was 3 - 2x + 6x. That would simply to 3 + 4x.

Now replace the x with &sqrt;(5). The radicand (the number under the radical) doesn't change in the same way that exponents don't change when you add terms with exponents.

The correct answer is (3).

6. Students were asked to write a polynomial given the following conditions:
• the degree of the expression is 3
• the leading coefficient is 2
• the constant term is -6
Which expression satisfies all three conditions?

(1) 4x - 6 + 3x²
(2) 3x² - 6x + 4
(3) 4 - 6x + 2x³
(4) 4x² + 2x³ - 6

Answer: (4) 4x² + 2x³ - 6

The degree of the polynomial is the highest exponent. The leading coefficient is the number in front of the term with the highest exponent because when the polynomial is in standard form, the term with the highest exponent is goes first (that is, it's the leading term). The constant, which usually goes last, doesn't have any variable.

The only polynomial that fits the conditions is Choice (4).

Choices (1) and (2) have degree 2. Choice (3) has a constant of 4.

7. Which graph below represents a function?

Answer: (1) [See Image]

A graph is a function so long as no x value has two corresponding y values. In other words, it will pass the Vertical Line Test in that no vertical line can be drawn that goes through two points.

Choice (1) has no points that line up vertically. This is the Correct choice.

Choice (2) has two pairs of vertical points. Eliminate Choice (2).

Choice (3) has two values when x = 4. Eliminate Choice (3).

Choice (4) has portions of three vertical lines. It fails the veritcal line test spectacularly. Eliminate Choice (4).

8. The following function models the value of a diamond ring, in dollars, t years after it is purchased:

v(t) = 500(1.08)^t

What was the original price of the ring, in dollars?

(1) $108
(2) $460
(3) $500
(4) $540

Answer: (3) $500

The original price happens when t=0. At t=0, v(t) = 500. This is Choice (3).
9. The formula for the surface area of a cylinder can be expressed as S = 2πr² + 2πrh, where r is the radius and h is the height of the cylinder. What is the height, h, expressed in terms of S, π, and r?

(1) h = (S - 2πr²) / (2πrh)
(2) h = S - r
(3) h = (2πr² - S) / (2πrh)
(4) h = r - S

Answer: (1) h = (S - 2πr²) / (2πrh)

Use inverse operations:

S = 2πr² + 2πrh

S - 2πr² = 2πrh

(S - 2πr²) / (2πr) = h

The correct choice is Choice (1).

10. When solving the following system of equations algebraically, Mason used the substitution method. 3x - y = 10
2x + 5y = 1
Which equation could he have used?

(1) 2(3x - 10) + 5x = 1
(2) 2(-3x + 10) + 5x = 1
(3) 2x + 5(3x - 10) = 1
(4) 2x + 5(-3x + 10) = 1

Answer: (3) 2x + 5(3x - 10) = 1

Changing 5y to 5x is just DUMB. Eliminate Choices (1) and (2).

Rewrite the first question to solve for y and then substitute for y in the second equation.

3x - y = 10
3x - 10 = y

The correction substitution is Choice (3).

11. Which graph represents the solution to the inequality 4 + 3x > 9 - 7x?

Answer: (3) [See Image]

The quick way to get this answer is to check x = 2 to see if the two expressions are equal. Then check x = 0:

If x = 2, then 4 + 6 = 10 and 9 - 14 = -5. Those aren't equal so eliminate Choice (1) and (2).

If x = 0, then 4 + 0 > 9 - 0 is NOT a true statement. Therefore x = 0 is NOT in the solution to the inequality. Eliminate Choice (4).

The correct choice is Choice (3).

Doing the work:

4 + 3x > 9 - 7x
10x > 5
x > 1/2

Choice (3) shows the graph of x > 1/2.

12. When solving the equation 3(2x + 5) - 8 = 7x + 10, the first step could be 3(2x + 5) = 7x + 18. Which property justifies this step?

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

Answer: (1) addition property of equality

In the first step, 8 was added to both sides of the equation. This is Choice (1).

13. Which table of values best models an exponential decay function?

Answer: (2) [See Image]

A decay function will decrease with a constant ratio. It will NOT have a constant rate of change.

Choice (1) decreases with a constant rate of 3. Eliminate Choice (1).

Choice (2) decreases by 20 then 18, then 16, then 15.
-20/200 = -.1, -18/180 = -.1, etc. The function is decreasing by 10% (and rounded to the nearest integer). This is the correct choiuce.

Choice (3) is increasing, so it isn't a decay function. Eliminate Choice (3).

Choice (4) decreases and then increases again, so it isn't a decay function. Eliminate Choice (4).

The correct choice is Choice (2).

14. If f(x) = √(x + 1) + 5, then what is the value of f(3)?

(1) 9
(2) 7
(3) 3
(4) 10

Answer: (2) 7

Because careful entering this into a calculator that you don't accidentally evaluate √(x + 1 + 5).

√(3 + 1) + 5 = √(4) + 5 = 2 + 5 = 7

The correct choice is Choice (2).

If you used √(x + 1 + 5), you would've gotten 3 as an answer and wouldn't have realized your error.

15. Isabella wants to shift the graph of the function f(x) = (x + 5)² - 2 left 3 units. Which function represents the shifted graph?

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

Answer: (2) g(x) = (x + 8)² - 2

The vertex is at (-5, -2). If it is shifted left 3 spaces, the vertex of the new function will be (-8,-2). The y-coordinate will not change.

This means that the new function will be g(x) = (x + 8)² - 2.

The correct choice is Choice (2).

16. What are the zeros of f(x) = x(x² - 36)?

(1) 0, only
(2) 6, only
(3) 6 and -6, only
(4) 0, 6, and -6

Answer: (4) 0, 6, and -6

If you factor x² - 36 into (x + 6)(x - 6), then you can see that the function has three terms with three distinct zeros, which are 0, -6 and 6.

Otherwise, you can graph the function, or you can substitute each of the three values to see what works.

The correct choice is Choice (4).

17. The point (x, -6) lies on the graph of a parabola whose equation is y = -x² - x + 6. The value of x can be

(1) -3 or 2
(2) -4 or 3
(3) 3, only
(4) -4, only

Answer: (2) -4 or 3

Graph the equation, or substitute each value until you get a correct answer.

Or you can solve for x:

x² - x + 6 = -6
x² - x + 12 = 0
(x - 4)(x + 3) = 0
x - 4 = 0 or x + 3 = 0
x = 4 or x = -3

The correct choice is Choice (2).

18. The two-way frequency table below is a summary of concession stand sales for a football game.

Of the people making a purchase at the concession stand, what is the relative frequency of them buying pizza and a water?

(1) 0.58
(2) 0.35
(3) 0.455
(4) 0.145

Answer: (4) 0.145

There were 58 people who bought water and pizza out of 400 people who made purchases.

To get the relative frequency, divide (but DO NOT make it a percentage):

58/400 = 0.145

The correct choice is Choice (4).

19. When Theodore was driving in Canada, his speed was 104 kilometers per hour. Theodore was asked to convert his metric speed to a different rate, using the following conversion:

Assuming he did all the work correctly, what would the units be for Theodore’s rate?

(1) feet per second
(2) feet per minute
(3) seconds per foot
(4) minutes per foot

Answer: (1) feet per second

Units can be multiplied and divided just like any other factor.

Cancel out any units that appear once in the numerator (top) and once in the denominator (bottom).

You'll get this:

That leaves feet per second.

The correct choice is Choice (1).

20. Which expression is equivalent to (-2x²)³?

(1) -2x⁵
(2) -2x⁶
(3) -8x⁵
(4) -8x⁶

Answer: (4) -8x⁶

(-2x²)³ is equivalent to (-2x²)(-2x²)(-2x²).

(-2)(-2)(-2) = -8, and (x²)(x²)(x²) = (x⁶)

The correct choice is Choice (4).

21. The table below shows the amount of a radiactive substance that remained for selected years.

To the nearest tenth, the average rate of change, in grams per year, from 2000 to 2014 is

(1) 39.1
(2) 51.8
(3) -39.1
(4) -51.9

Answer: (4) -51.9

Ignore every column except 2000 and 2014. Find the rate of change for those two points.

First off, the rate of change is negative, so Eliminate Choices (1) and (2).

Calculate (25 - 750) / (2014 - 2000) = -725/14, or about 51.8.

The correct choice is Choice (4).

22. When 2x² -3x + 4 is subtracted from x² + 2x - 5, the result is

(1) x² - 5x + 9
(2) x² - x + 1
(3) -x² + 5x - 9
(4) -x² - x - 1

Answer: (3) $500

When subtracting the expression AFTER the "from" goes first.

x² + 2x - 5
2x² -3x + 4
-x² + 5x - 9

The correct choice is Choice (3).

23. Which equation has the same solution as x² - 6x = 24?

(1) (x - 3)² = 24
(2) (x - 6)² = 24
(3) (x - 3)² = 33
(4) (x - 6)² = 60

Answer: (3) (x - 3)² = 33

When Completing the Square, the number in the parentheses will be half the amount of the coefficient of the x term in the original equation. Eliminate Choices (2) and (4).

It should be "obvious" that Choice (1) is incorrect, but just in case, let's work it out:

Half of -6 is -3, and -3 squared is +9, so 9 must be added to both sides of the equation.

x² - 6x = 24

x² - 6x + 9 = 24 + 9

(x - 3)² = 33

The correct choice is Choice (3).

24. In a sequence, the first term is -2 and the common ratio is -3. The fourth term in this sequence is

(1) -162
(2) -11
(3) 24
(4) 54

Answer: (4) 54

You only need the fourth term, so skip the formulas and just work out the first four terms. Start with -2 and multiply each term by -3 to get the next term.

-2, 6, -18, 54.

The fourth term is 54. (The fifth term would be -162.)

The correct choice is Choice (4).

More to come. Comments and questions welcome.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! If you enjoy my books, please consider leaving a rating or review on Amazon or on Good Reads. Thank you!

Allergies

(Click on the comic if you can't see the full image.)

(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Based on a true story.

I've actually worked on a few comic for use in my classroom over the past few months, and I'll rework them for use on this page. I've been teaching a Graphic Novel course that is an Art credit even though I'm neither Art nor English teacher. We're learning about them and their history, and trying to create some of our own, without worrying to much about the actual artwork. It's more about the effort and design we're putting into them. For anyone really interested, developing art skills (some already have them) will come later. Or they can tell their story by partnering with an artist, and they'll be able to explain exactly what they envision.

Much of what follows appears on the page from the last comic I published -- except that I published it quite a while after the date on the comic.

The slightly longer story: every time I looked at the blog after that, the old "imposter syndrome" reared its ugly head. And then there's the joke asking if you could still be paranoid if people really are out to get you, and I wondered if that applied to with this syndrome as well. It's not like I ever achieved major success with this blog, and for the time that I did, I really didn't know how to capitalize on that to take it to another level (like, for example, Math with Bad Drawings).

And now blogs are past their heyday. Even now, my Regents pages, which always got more hits than my comics, are down in views, as search engine AI spit out answers before they even point to my pages. Fewer of today's high schoolers would even know I exist.

So that's what's been going on. I've been in a decline pretty much since I knew I was closing in on my 2,000th strip and I would need to do something for it. I think I'm past the point of doing something BIG, not because no one is expecting much, but because few seem to be out there at all. And I did MS Paint myself into a corner with a breakup story that was supposed to be resolved fairly quickly -- but I had to write it and then edit down what I was thinking, and then draw it. I hope I will get to that because I don't want to leave those characters hanging. (They could eventually reconcile off-screen.)

If you are out there, I will remind you of the many times that I've said, "I thrive on feedback." And then I might point out that I get so little of it.

MY NEWEST BOOK IS OUT Burke's Lore Briefs: Yesterday's Villains, the following to Tomorrow's Heroes is now available on Amazon and Kindle Unlimited. If Heroes who don't die live long enough to become the villain, what happens to Villains who live long enough? When do schemes of global conquest become dreams of a quiet place away from all those annoying people you once wanted to subjugate? And does anyone really want to rule over the world's ashes if it means we can't have nice things?
My older books include three more books in my Burke's Lore Briefs series, and the anthologies A Bucket Full of Moonlight and In A Flash 2020. Vampires, werewolves, angels, demons, used-car salesmen, fairies, superheroes, space and time travel, and little gray aliens talking to rock creatures and living plants. Plus pirates, spies, horror, and kindergarten noir! 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.


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.
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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!

Geometry Problems of the Day (Geometry Regents, August 2025 Part IV)

This exam was adminstered in August 2025.

August 2025 Geometry, Part IV

Each correct answer is worth up to 6 credits. Partial credit can be given. Work must be shown or explained.

35. In quadrilateral ABCD below, side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC.

Prove ABCD is a parallelogram.

Answer:

The rule you need to remember here is that a quadrilateral with one pair of opposite sides that are both parallel and congruent is a parallelogram.

We can prove this by showing that the two triangles in the image are congruent. This will show that the AB || CD and then the transitive property will show that AB ≅ CD.

Statement	Reason
1. quadrilateral ABCD with side CD is extended through D to point E such that AFD and BFE bisect each other, and DE ≅ DC.	Given.
2. AF ≅ FD and EF ≅ FB.	Definition of bisector.
3. ∠AFB ≅ ∠DFE	Vertical angles are congruent.
4. △DFE ≅ △AFB	SAS Postulate
5. ∠DEF ≅ ∠ABF	CPCTC
6. AB || EC	If two lines are cut by a transversal and the Alternate Interior Angles are congruent, then the lines are paralllel.
7. AB ≅ DE	CPCTC
8. AB ≅ DC	Transitive Property / Substitution
9. ABCD is a parallelogram	In a quadrilateral, if two opposite sides are parallel and congruent then the quadrilateral in a parallelogram.

One curious note:
In Step 8, I would've stated the reason as "the transitive property". It's curious that in the all the correct solutions in the model response set, the word "substitution" is used. In fact, there's only one example that uses the phrase "Transitive Property", but it uses it incorrectly -- there was a step missing before that. This has me questioning if that is a sufficient "Reason" for full credit. I like it better than "substitution" myself, which is also logical, but sounds more algebraic to me.

End of Part Exam

How did you do?

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!