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.

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