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

The Problems of the Day are BACK! Let's see for how long. Life, y'know?

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.


17. The formula for the area of a trapezoid is A = (1/2)h(b₁ + b₂). The height, h, of the trapezoid may be expressed as

(1) 2A / (b₁ + b₂)
(2) (1/2)A(b₁ + b₂)
(3) (b₁ + b₂) / (2A)
(4) (1/2)A - (b₁ + b₂)

Answer: (1) 2A / (b₁ + b₂)

Inverse operations to isolate h in terms of the other variables.

A = (1/2)h(b₁ + b₂)
2A = h(b₁ + b₂)
2A / (b₁ + b₂) = h

Choice (1) is the correct answer.

10. Three functions are given below.

Which functions have the same y-intercept?

(1) f(x) and g(x)
(2) g(x) and h(x
(3) f(x) and h(x)
(4) The functions all have different y-intercepts.

Answer: (1) f(x) and g(x)

The y-intercept of h(x) is in the table -- when x = 0, h(x) = -3. Now calculate f(0) and g(0), or graph them.

f(0) = -|0 + 2| + 7 = -2 + 7 = 5, so f(x) and h(x) have different y-intercepts.

g(0) = (0 - 3)² - 4 = 9 - 4 = 5, which is the same as f(0).

Therefore, the correct answer is Choice (1).

19. The sum of (x + 7)² and (x - 3)² is

(1) 2x² + 58
(2) 2x⁴ + 58
(3) 2x² + 8x + 58
(4) 2x⁴ + 8x² + 58

Answer: (2) -3.75

The sum of two quadratics is also a quadratic (unless the x² terms cancel out for some reason, in which case the order of lower, but it'll never be higher.) Eliminate Choices (2) and (4).

It should also be obvious that Choice (1) is incorrect because there is no reason why the middle term should be missing.

(x + 7)² + (x - 3)²

x² + 14x + 49 + x² - 6x + 9

2x² + 8x + 58

Choice (3) is the correct answer.

20. The product of 2√(10) and 3√(2) is

(1) 12√(5)
(2) 5√(20)
(3) 24√(5)
(4) 5√(12)

Answer: (1) 12√(5)

Multiply the coefficients and multiply the radicands.

So (2√(10)) * (3√(2)) = 6√(20), which is not one of the choices. However, the radical can be simplified.

6√(20) = 6√( (4)(5) ) = 6√(4)&sqrt;(5) = 12√(5)

This is Choice (1).

This problem could've been solved by putting all four choices and the original question into a calculator.

21. When 6x³ - 2x + 8 is subtracted from 5x³ + 3x - 4, the result is

(1) x³ - 5x + 12
(2) x³ + x + 4
(3) -x³ + 5x - 12
(4) -x³ + x + 4

Answer: (3) -x³ + 5x - 12

The most important word in this problem is FROM. The 5x³ polynomial is on top (or on the left) and the 3x³ polynomial is on the bottom (or on the right).

Next, be careful with subtracting signed numbers.

(5x³ + 3x - 4) - (6x³ - 2x + 8) = -x³ + 5x - 12, which is Choice (3).

22. Three relations are shown below

Which relations represent a function?
(1) I and II, only
(2) I and III, only
(3) II and III, only
(4) I, II, and III

Answer: (4) I, II, and III

A relation is a function if none of the x-values repeat.

In function I, the domain is {0,1,2,3}, which makes I a function.

In function II, the domain is {3,4,5,6}, and no member of the domain has two arrows coming outwhich makes II a function.

In function III, the graph passes the vertical line test because every closed circle is over an open circle, so III is a function.

The correct answer in Choice (4).

23. The method of substitution was used to solve the system of equations below:

4x - 7y = 7
x - y = -1

Which equation is a correct first step when using this method?

(1) x = y - 1
(2) y = x - 1
(3) 3x - 6y = 8
(4) 5x - 8y = 6

Answer: (1) x = y - 1

This can be solved using substitution. The first step would be to solve for x or solve for y.

Given x - y = -1, then x = y - 1 and y = x + 1.

This means that Choice (1) is correct and Choice (2) can be eliminated.

Choices (3) and (4) are found by subtracting or adding the two equations. Neither eliminates a variable, so it isn't helpful.

Note that you can subtract the second equation four times to eliminate the x term, but that would not be using Substitution.

24. In 2009, Usain Bolt, a sprinter from Jamaica, set the world record in the 100-meter dash with a time of 9.58 seconds. His approximate speed, in kilometers per hour, can be found using which conversion?

Answer: (4) [SEE IMAGE]

I love using Usain Bolt references in class, particularly in semesters when I have Jamaican students.

Units are like variables in that they can be multiplied and divided. That is, feet times feet are feet² or square feet. And miles/hour times hours gives you miles because hours divided by hours is 1.

Look at the four choices and cancel out the units that appear in both the top and bottom of the fraction.

Here is what the four choices should look like:

In Choice (1), we have hr/km not km/hr. Eliminate Choice (1).

In Choice (2), we have m²/(km*hr). Eliminate Choice (2).

In Choice (3), we have (km*hr)/sec². Eliminate Choice (3).

In Choice (4), we have km/hr. This 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!

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

The Problems of the Day are BACK! Let's see for how long. Life, y'know?

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.


9. The tables below show the input and output values of four different functions.

(1) f(x)
(2) g(x)
(3) h(x)
(4) j(x)

Answer: (4) j(x)

The output values repeat in f(x) and g(x), so eliminate Choices (1) and (2) as they change direction and can't be linear.

The rate of change in h(x) is changing, not constant. It goes down 1, down 2, down 4, etc. Eliminate Choice (3).

The rate of change in j(x) is constant. It goes up 4 from -11 to -7, then up 4 again to -3, etc. This is the correct choice.

10. What is the solution set to the equation 3x² = 24x?

(1) {8}
(2) {0,8}
(3) {0,-8}
(4) {0,8,-8}

Answer: (2) {0,8}

You can solve this by factoring:

3x² = 24x
x² = 8x
x² - 8x = 0
x(x - 8) = 0
x = 0 or x - 8 = 0

x = 0 or x = 8

A quadratic equation can have at most two solutions, so Choice (4) could've been eliminated immediately.

Likewise, since x = 0 is "obviously" a solution, Choice (1) could have been eliminated as well.

If you didn't wish to factor, you could've checked 0, 8, and -8 in a calculator.

11. The table below shows the radioactivity level of a substance after the given time, t, in seconds.

What is the average rate of change in radioactivity level over the interval 1 < t < 3?

(1) 3.75
(2) -3.75
(3) 4.6875
(4) -4.6875

Answer: (2) -3.75

Average rate of change is the slope of the line from t = 1 to t = 3.

(2.5 - 10) / (3 - 1) = -7.5 / 2 = -3.75

That is Choice (2).

12. Fred recorded the number of minutes he read each day, from Monday through Friday. His results are shown in the table.

What is the correlation coefficient, to the nearest thousandth, and strength of the linear model of these data?

(1) 0.984 and strong
(2) 0.968 and strong
(3) 0.984 and weak
(4) 0.968 and weak

Answer: (1) 0.984 and strong

This is an annoying question for the multiple-choice portion of the exam because of the work involved. You can't reason out the final answer because 0.984 and 0.968 are so close together. There's no short cut.

However, both 0.984 and 0.968 indicate strong relationship, so Choices (3) and (4) can be eliminated.

You need to put the data into Lists in your graphing calculation and run a Linear Regression.

You will then find that r² = 0.968 and r = 0.984. You want the r value. (You'll never want the r² value for this level of algebra.

The correct answer is Choice (1).

13. Given f(x) = x², which function will shift f(x) to the left 3 units?

(1) g(x) = x² + 3
(2) h(x) = x² - 3
(3) j(x) = (x - 3)²
(4) k(x) = (x + 3)²

Answer: (4) k(x) = (x + 3)²

If you don't remember the rules, you can graph each of these and check the graphs or Tables of Values.

In Choice (1), the "+ 3" shifts the graph up 3 units. Eliminate Choice (1).

In Choice (2), the "- 3" shifts the graph down 3 units. Eliminate Choice (2).

In Choice (3), the "- 3" inside the parentheses shifts the graph to the right 3 units. Eliminate Choice (3).

In Choice (4), the "+ 3" inside the parentheses shifts the graph to the left 3 units. Choice (4) is the correct answer.

14. A class of 20 students was surveyed to determine the number of pets each student owned. The data are represented in the dot plot below.

Which statement about the data is correct?
(1) The mean and the median are the same.
(2) The median and the mode are the same.
(3) The mean and the mode are the same.
(4) The mean, median, and mode are all the same.

Answer: (2) The median and the mode are the same.

The mode is 2 because it occurs the most. The median is 2 because there are 20 students and both the 10th and 11th numbers are 2.

Choice (2) says "the median and the mode are the same", which is true. Unfortunately, Choice (4) says all three are the same, so you still have to calculate the mean.

The mean is the Sum / 20, which is 50 / 20 = 2.5. So Choices (1), (3), and (4) are incorrect.

15. The range of f(x) = |x + 2| - 5 is

(1) y > -5
(2) y > 2
(3) x > -5
(4) x > 2

Answer: (1) y > -5

Range deals with the y values, so eliminate Choices (3) and (4).

The vertex of this absolute value function is (-2,-5). That is the lowest point of the function. Therefore, the range is every value of y that is greater than -5. That is Choice (1).

16. Which equation is always correct?

(1) a³ • a^x = a^3x
(2) (a⁴)^x = a^{4 + x}
(3) (ab)^x = a^xb^x
(4) a^x • b^y = ab^{x + y}

Answer: (3) (ab)^x = a^xb^x

These are illustrating the laws of exponents, but only one of them is correct.

If you don't remember the rules, pick some prime numbers like a = 2, b = 3, x = 5 and y = 7 and check which are correct. If more than one choice is correct, pick different numbers and check again.

In Choice (1), a³ • a^x = a^{3 + x} not a^3x. Eliminate Choice (1).

In Choice (2), (a⁴)^x = a^4x not a^{4 + x} . Eliminate Choice (2).

In Choice (3), (ab)^x = a^xb^x. This is the correct answer. Both a and b will get the exponent.

In Choice (4), a^x • b^y cannot be combined, and ab^{x + y} is just crazy. (For one thing, the exponent only applies to b while a no longer has any exponent.) Eliminate 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!

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.

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 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: (4) [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!

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.


9. When the formula p = 2l + 2w is solved for w, the result is

(1) w = (2l + p) / 2
(2) w = (p - 2l) / 2
(3) w = p/2 + l
(4) w = l - p/2

Answer:(2) w = (p - 2l) / 2

Use inverse operations to isolate w.

p = 2l + 2w

p - 2l = 2w

(p - 2l) = w

The correct answer is Choice (2).

10. Market Street Pizza kept a record of pizza sales for the month of February. The results are shown in the table below.

Of all the pizzas sold in February, what percent were plain, deep-dish pizzas?

(1) 20%
(2) 30%
(3) 40%
(4) 50%

Answer: (1) 20%

There were 200 plain, deep-dish pizzas sold. The percentage would be 200 divided by the total number of pizzas sold. Find the grand total of all pizzas in the table.

300 + 200 + 80 + 25 + 120 + 105 + 100 + 70 = 1000 pizzas sold.

200/1000 = 0.20 = 20%, which is Choice (1).

11. When solving -2(3x - 5) = 9/2 x - 2 for x, the solution is

(1) 8/7
(2) 10/11
(3) -16/21
(4) -16/3

Answer: (1) 8/7

You can solve this with the calculator but be sure to use parentheses when substituting.

-2(3x - 5) = 9/2 x - 2

-6x + 10 = 9/2 x - 2

-12/2 x + 10 = 9/2 x - 2

-21/2 x = -12

-21x = -24

x = -24/-21

x = 8/7

The correct Choice is Choice (1).

12.The expression x^{2a + b} is equivalent to

(1) x^2a + x^b
(2) x^a + x^{a + b}
(3) x^a * x^{a + b}
(4) x^{a + b} * x^{a + b}

Answer: (3) x^a * x^{a + b}

Terms with different exponents CANNOT be combined because they are not Like Terms. Terms with the same base can be multiplied. When multiplying terms with the same base, the exponents are added.

The first two choices cannot be added together because they are not like terms. Eliminate Choices (1) and (2).

In Choice (3), x^a * x^{a + b} = x^{2a + b}, which is the correct answer.

In Choice (4), x^{a + b} * x^{a + b} = x^{2a + 2b}. Eliminate Choice (4).

13. The inputs and outputs of a function are shown in the table below.

This function can best be described as

(1) linear
(2) quadratic
(3) exponential
(4) absolute value

Answer: (3) exponential

There is no common difference but there is a common ratio. You can see from the bottom three lines that 0.5 times 2 is 1 and 1 times 2 is 2. You can check the others as well. Going up the table, each decimal is half as large as the one below it.

Since the table has a common ratio, it is exponential, which is Choice (3).

14. Stephanie is solving the equation x² - 12 = 7x - 8. Her first step is shown below. Given: x² - 12 = 7x - 8
Step 1: x² - 4 = 7x

(1) associative property
(2) commutiative property
(3) distributive property
(4) addition property of equality

Answer: (4) addition property of equality

To get from the Given equation to the Step 1, Stephanie added 8 to both sides of the equation. This is possible because of the addition property of equality.

The correct answer is Choice (4).

15. What is the sum of 8√(3) and √(3)?

(1) 8√(6)
(2) 9√(6)
(3) 7√(3)
(4) 9√(3)

Answer: (4) 9√(3)

The sum of 8x and x is 9x. This is true whether x is apples, kittens, -27, or √(3).

Choice (4) is the correct answer.

16. The dot plots below represent test scores for 20 students on a math test.

The mode for this math test is 80 and the median is 85. Which dot plot correctly represents this data?

(1) I
(2) II
(3) III
(4) IV

Answer: (1) I

The mode of a dot plot is the number with the highest columns of dots. That eliminates Choices (2) and (4).

The median of 20 students is the average of the 10th and 11th score. In plot I, the 10th and 11th dots are both 85. In plot III, the 10th and 11th dots are in the 80 column.

Eliminate Choice (3).

Choice (1) is the correct answer.

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?
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