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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