Algebra Problems of the Day (Algebra Regents, August 2025 Part I)

This exam was adminstered in August 2025.

More Regents problems.

August 2025 Algebra Regents

Part I

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


1. Which expression is equivalent to 100x² − 16?

(1) (50x - 8)(50x + 8)
(2) (50x - 8)(50x - 8)
(3) (10x - 4)(10x + 4)
(4) (10x - 4)(10x - 4)

Answer: (3) (10x - 4)(10x + 4)

This is the Difference of Two Perfect Squares Rule. Take the square root of each term and write the conjagtes -- that is, write two binomials that are the same except that one is separated by addition (+) and one is separated by subtraction (-).

In Choices (1) and (2), 50x times 50x is 2500x², not 100x². Eliminate these.

In Choice (4), you are squaring (10x - 4), which will produce a constant of + 16, not - 16, as well as a middle term of -80x. Eliminate Choice (4).

Choice (3) is the Correct answer.

2. Josie has $2.30 in dimes and quarters. She has two more dimes than quarters. Which equation below can be used to determine x, the number of quarters she has?

(1) 0.35(2x + 2) = 2.30
(2) 0.25(x + 2) + 0.10x = 2.30
(3) 0.25x + 0.10(x + 2) = 2.30
(4) 0.25x + 0.10(x − 2) = 2.30

Answer: (3) 0.25x + 0.10(x + 2) = 2.30

If x is the number of quarters, then 0.25x is the amount that those quarters are worth. Eliminate Choices (1) and (2).

If there are 2 more dimes than quarters, then the number of dimes is x + 2, not x - 2. So the correct answer is Choice (3).

3. If g(x) = −2x² + 16 then g(−3) equals

(1) -20
(2) -2
(3) 34
(4) 52

Answer: (2) -2

Substitute and evaluate. Put -2(-3)² + 16 into your calculator.

g(-3) = -2(9) + 16 = -18 + 16 = -2, which is Choice (2).

4. What are the zeros of f(x) = x² − 8x − 20?

(1) 10 and 2
(2) 10 and -2
(3) -10 and 2
(4) -10 and -2

Answer: (2) 10 and -2

Since the constant term is negative, then one of the roots will be positive and the other will be negative. Eliminate Choices (1) and (4).

You can factor and solve the quadratic equation, or you can try the choices to see what works.

In Choice (2), (10)² - 8(10) - 20 = 100 - 80 - 20 = 0 is a true statement. Since we've already eliminated Choice (1), Choice (2) must be the correct answer.

You can check the other root: (-2)² - 8(-2) - 20 = 4 + 16 - 20 = 0, which is true.

Algebraically:

x² − 8x − 20 = 0
(x - 10)(x + 2) = 0
x - 10 = 0 or x + 2 = 0
x = 10 or x = -2

5. Which point lies on the graph of y = 3x² - 1/4 x + 3?

(1) (−2,15.5)
(2) (−1,5.75)
(3) (1,6.25)
(4) (2,15.5)

Answer: (1) (−2,15.5)

Check each x in the calculator to see if you get the correct y value. Or graph the function and look at the Table of Values.

In Choice (1), 3(-2)² - 1/4(-2) + 3 = 15.5, which is correct. Choice (1) is the correct choice.

In Choice (2), 3(-1)² - 1/4(-1) + 3 = 6.25, not 5.75. Eliminate Choice (2).

In Choice (3), 3(1)² - 1/4(1) + 3 = 5.75, not 6.25. Eliminate Choice (3).

In Choice (2), 3(2)² - 1/4(2) + 3 = 14.5, not 15.5. Eliminate Choice (4).

6. Given f(x) = x² and g(x) = 8x − 15 graphed on the same set of axes, which value(s) of x will make f(x) = g(x)?

(1) 3, only
(2) 9, only
(3) 3 and 5
(4) 9 and 25

Answer: (3) 3 and 5

Graph both equations and check the Table of Values for 3, 5, 9 and 15. Or substitute and solve.

f(3) = 3² = 9 and g(3) = 8(3) - 15 = 24 - 15 = 9, so f(3) = g(3). This means the correct answer is either (1) or (3). Eliminate Choice (2) and (4).

f(5) = 5² = 25 and g(5) = 8(5) - 15 = 40 - 15 = 25, so f(5) = g(5). The correct answer is Choice (3).

Algebraically:

f(x) = g(x)
x² = 8x - 15
x² - 8x + 15 = 0
(x - 3)(x - 5) = 0
x - 3 = 0 or x - 5 = 0
x = 3 or x = 5

7. Which trinomial is written in standard form and has a constant term of five?

(1) x⁵ − 4x² + 10
(2) 2x² + 6x⁴ + 5
(3) 5x⁴ − 3x² + 1
(4) 4x⁵ − 8x² + 5

Answer: (4) 4x⁵ − 8x² + 5

A constant is a numerical term without a variable. Choice (1) has a constant of 10, and Choice (3) has a constant of 1. Eliminate them.

Choice (2) is not in standard form because the term with the highest exponent is not first. Eliminate Choice (2).

Choice (4) is the correct answer.

8. When solving x² + 6x = −8 for x, a student wrote x² + 6x + 8 = 0 as their first step. Which property justifies this step?

(1) associative property
(2) commutative property
(3) zero property of addition
(4) addition property of equality

Answer: (4) addition property of equality

Adding 8 to both sides of the equation is an example of the addition property of equality. This is Choice (4).
More to come. Comments and questions welcome.

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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 Problems of the Day (Algebra Regents, January 2025 Part I)

This exam was adminstered in January 2025.

More Regents problems.

January 2025 Algebra Regents

Part I

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


17. A function is graphed on the set of axes below.

The domain of the function is

(1) {x | x > -2}
(2) {x | x > -2}
(3) {x | x > -4}
(4) {x | x > -4}

Answer: (1) {x | x > -2}

The open circle indicates greater than but not equal to. Eliminate Choices (2) and (4).

The domain is the x value and the lowest possible x value is -2. The correct answer is Choice (1).

Choices (3) and (4) confuse the domain with the range, except they use x instead of y. The range is {y | y > -4}.

18. Which ordered pair is a solution to the equation y − 1 = 2(x + 1/4)?

(1) (0.75,0)
(2) (1.25,4)
(3) (2.5,-6.5)
(4) (4,-9.5)

Answer: (2) (1.25,4)

Add 1 to both sides of the equation and check y = 2(x + .25) + 1 for each of the four choices.

Choice (1), y = 2(.75 + .25) + 1 = 3, not 0. Eliminate Choice (1).

Choice (2), y = 2(1.25 + .25) + 1 = 4. Choice (2) is the correct answer.

In Choices (3) and (4), it should be obvious that the value of y for the given x will be positive and not negative, so both Choices (3) and (4) could be eliminated. However, if you didn't notice that, we can continue.

Choice (3), y = 2(2.5 + .25) + 1 = 6.5, not -6.5. Eliminate Choice (3).

Choice (43), y = 2(4 + .25) + 1 = 9.5, not -9.5. Eliminate Choice (4).

19. Elena’s fastest time for the 50-meter dash is 7 seconds. She wants to know how fast this is in inches per minute. Which expression can Elena use for a correct conversion?

Answer: (4) [See Image]

If you are changing meters per second into inches per minute than inches needs to appear once on top and minutes need to appear once on the bottom. The starting fraction needs to be meters/second, and both of those units need to be cancelled.

Choices (1) and (2) start with seconds per meter. Eliminate both of them.

Choice (3) has meters twice in the numerator, so neither one cancels. The end result will be meters² per min-in. Eliminate Choice (3).

Choice (4) has meters once in the numerator and once in the denominator, which cancel. It has seconds once in the numerator and once in the denominator, which cancel. The final result is in/meter. Choice (4) is the correct answer.

20. The table below shows the highest temperatures recorded in August for several years in one town.

The interquartile range of these data is

(1) 7
(2) 10
(3) 11
(4) 18

Answer: (3) 11

You can put the temperatures (not the years) into a list in your calculator and get a Five-Number Summary by doing 1-Var Stats.

Or you could arrange the data in order and do it yourself:

77, 78, 81, 84, 86, 88, 93, 95

There are 8 pieces of data, so the average of the fourth and fifth is the median, which is 85, but not important to this problem.

The median of the first four numbers is 79.5, which is Q1. The median of the last four numbers is 90.5, which is Q3. The IQR is Q3 - Q1, which is 90.5 - 79.5, which is 11. This is Choice (3).

21. The function f(x) = x² is multiplied by k, where k < −1. Which graph could represent g(x) = kf(x)?

Answer: (4) [See Image]

Multiplying a quadratic function by a factor greater than 1 will make it skinnier. Multiplying a quadratic function by a factor less than -1 will make it skinnier AND flip the direction of the parabola.

The given function, f(x) = x², opens upward. The new function, g(x), will open downward and be thinner than the original. This is Choice (4).

If 0 < k *lt 1, then Choice (1) would've been the correct answer.

If -1 < k < 0, then Choice (2) would've been the correct answer.

If k > 1, then Choice (3) would've been the correct answer.

22. Which graph is the solution to the inequality 6.4 − 4x ≥ −2.8?

Answer: (2) [See Image]

Since greater than or equal to is specified, the answer will have a closed circle. Eliminate Choices (1) and (3).

Next, you can check to see if x = 0 is in the solution set or not.

6.4 - 4(0) ≥ −2.8?
6.4 ≥ −2.8 (check)

Since 0 is in the solution set, the correct choice must be Choice (4) and not Choice (2).

Algebraically,

6.4 - 4x ≥ −2.8
-4x ≥ −9.2
x < 2.3

Remember that when you divide (or multiply) an inequality by a negative number, you have to flip the direction of the inequality symbol.

The correct answer is Choice (2).

23. The number of fish in a pond is eight more than the number of frogs. The total number of fish and frogs in the pond is at least 20. If x represents the number of frogs, which inequality can be used to represent this situation?

(1) x + 8x ≥ 20
(2) 2x + 8 ≥ 20
(3) x + 8x ≤ 20
(4) 2x + 8 ≤ 20

Answer: (2) 2x + 8 ≥ 20

The term 8x refers to 8 times as many fish as frogs, not 8 more than the number of frogs. Eliminate Choices (1) and (3).

The pond has at least 20 fish and frogs meaning that the combined number is greater than 20. The correct choice is Choice (2).

24. Which graph below represents a function that is always decreasing over the entire interval −3 < x < 3?

Answer: (4) [See image]

Always decreasing means as x gets bigger, y gets smaller. The graph is always going down and to the right.

Choice (1) is increasing in the interval −3 < x < 3. Eliminate Choice (1).

Choice (2) is increasing in the interval −3 < 0 and then starts decreasing. Eliminate Choice (2).

Choice (3) is increasing in the interval −3 < x < 3. Eliminate Choice (3).

Choice (4) is the correct answer.

End of Part I. 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!