This exam was adminstered in June 2026.
More Regents problems.
June 2026 Algebra Regents
Part I
Each correct answer will receive 2 credits. No partial credit.
1. What is the 20th term of the arithmetic sequence 4, 7, 10, 13, ...?
(1) 61
(2) 64
(3) 79
(4) 83
Answer: (1) 61
The Common Difference is 7 - 4 = 3. The 20th term is 19 X 3 more than the first term.
4 + 19 X 3 = 61, which is Choice (1).
2. What is the value of x in the equation 0.5x − 4 = 8 − x?
(1) 6
(2) 8
(3) 18
(4) 24
Answer: (2) 8
Isolate the variable by adding x to both sides of the equation and then use inverse operations.
0.5x − 4 = 8 − x
1.5x − 4 = 8
1.5x = 12
x = 8
Choice (2) is the correct answer.
3.The binomial 4x2 − 25 is equivalent to
(1) 4(x+5)(x−5)
(2) 4(x−5)(x−5)
(3) (2x−5)(2x+5)
(4) (2x−5)(2x−5)
Answer: (3) (2x−5)(2x+5)
The Difference of Squares Rule applies. Every part of this binomial is a perfect square, so the binomial can be broken down into two conjugates.
4x2 − 25 = (2x - 5)(2x + 5), which is Choice (3).
4. A function is defined by the following set of points:
What is a possible value for x?
(1) 1
(2) 2
(3) 3
(4) -4
Answer: (2) 2
For a relation to be a function, no input value can repeat with a different output value. The numbers 1, 3, and -4 are already in this function and can be eliminated.
That leave only Choice (2) 2, which is the correct answer.
The fact that 2 appears in the output doesn't matter.
5. The expression (4xy2)3 is equivalent to
(1) 12x3y6
(2) 12x3y8
(3) 64x3y6
(4) 64x3y8
Answer: (3) 64x3y6
The exponent of 3 is applied to every coefficient and variable inside the parentheses. The exponents are multiplied. If they don't have an exponent, then the exponent is 1.
(4xy2)3 = 43x3y(2*3) = 64x3y6
This is Choice (3).
6. Allison was asked to write a third-degree trinomial with a leading coefficient of 4 and a constant term of 5. Which expression satisfies these conditions?
(1) 4x3 - 5
(2) 3x4 + 5
(3) 4x3 + 8x2 + 5
(4) 3x4 + 2x3 - 5
Answer: (3) 4x3 + 8x2 + 5
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 (3).
Note: This is almost EXACTLY the question that was asked in January for question number 6.
Choice (1) has a constant of -5.
Choices (2) and (4) have degree 4 and a leading coefficient of 3.
7. Given the sequence 128,64,32,..., which formula could be used to find the nth term of this sequence?
(1) an = 128(-2)n - 1
(2) an = 128(1/2)n - 1
(3) an = 128 - 2(n - 1)
(4) an = 128 + (1/2)(n - 1)
Answer: (2) an = 128(1/2)n - 1
Each term is one half of the previous term. The Common Ratio is 1/2. This is a geometric sequence. Eliminate Choices (3) and (4).
In Choice (1), the sign would be changing, and every other term would be four times greater than the term two before it. Eliminate Choice (1).
Choice (2) is the Correct choice.
8. When a bicyclist increases the pressure on the brakes, the speed of the bicycle decreases. This relationship can best be described as a
(1) negative correlation and causal relationship
(2) negative correlation and non-causal relationship
(3) positive correlation and causal relationship
(4) positive correlation and non-causal relationship
Answer: (1) negative correlation and causal relationship
The brakes slow the bike. The more pressure on the brakes, the slower the speed of the bike.
The correlation is causal and negative.
Note that the word is "causal" and NOT "casual". If you misread this, you may be somewhat confused because that doesn't mean anything.
9. A garden club plans to plant 40 flowering plants this year. They will only purchase daffodils that cost $4 per plant and tulips that cost $5 per plant. All prices include tax. The club has $170 to spend on plants. Which equation could be used to find the number of daffodil plants, d, the club purchases?
(1) 4d + 5(170 − d) = 40
(2) 4d + 5(40 − d) = 170
(3) 5d + 4(170 − d) = 40
(4) 5d + 4(40 − d) = 170
Answer: (2) 4d + 5(40 − d) = 170
The total is $170, so eliminate Choices (1) and (3).
Daffodils cost $4, so the cost of all the daffodils bought will be 4d. Eliminate Choice (1).
The correct choice is Choice (2).
A system of equations could be 4d + 5t = 170 and d + t = 40, so t = 40 - d, which could be substituted into the first equation.
10. A function is defined as h(x) = x2 − 3x + 1. What is the value of h(−1)?
(1) 1
(2) 2
(3) 3
(4) 5
Answer: (4) 5
Substitute and evaluate. Make sure you use parentheses around the -1.
h(-1) = (-1)2 − 3(-1) + 1 = 1 + 3 + 1 = 5
The correction substitution is Choice (4).
11. The zeros of p(x) = x(3x + 2)(x − 5) are BR>
(1) -2/3 and 5, only
(2) 2/3 and -5, only
(3) -2/3, 0, 5
(4) 2/3, 0, -5
Answer: (3) -2/3, 0, 5
There are three distinct linear factors, so there will be three distinct zeroes. Eliminate Choices (1) and (2).
You could graph the function and look at the table of values (or the graph for that matter).
If x - 5 = 0, then x = 5, this is in Choice (3).
If 3x + 2 = 0, then 3x = -2 and x = -2/3, which is also Choice (3).
The correct choice is Choice (3).
12. If f(x) = 1.25x and g(x) = 3x + 10, what is the smallest positive integer of x for which f(x) > g(x)?
(1) 18
(2) 19
(3) 67
(4) 69
Answer: (2) 19
You could graph the function and look at the table of values.
f(18) = 55.511..., g(18) = 64
f(19) = 69.388..., g(19) = 67
Choice (2) is the correct answer.
Note that the values of g(19) and f(19) are Choices (3) and (4), respectively.
13. What is an equation of the line that passes through the points (2,5) and (−2,−1)?
(1) y - 5 = 2/3 (x - 2)
(2) y - 5 = 3/2 (x - 2)
(3) y - 2 = 2/3 (x - 5)
(4) y - 2 = 3/2 (x - 5)
Answer: (2) y - 5 = 3/2 (x - 2)
The formula is y - y1 = m(x - x1) -- or sub 0, if you prefer. Either way, for the point (2,5), x = 2 and y = 5. Eliminate Choices (3) and (4).
Calculate the slope: m = (-1 - 5) / (-2 - 2) = (-6)/(-4) = 3/2. So Choice (2) is the correct answer.
14. At a local high school, students were asked to name the sport they like to watch the most. The results are summarized in the table below.
Approximately what percentage of female preferred to watch basketball?
(1) 35
(2) 40
(3) 53
(4) 57
Answer: (4) 57
You only care about the "Female" row of the table, but you need to find the total number of Females who responded.
THere were 40 who liked to watch basketball out of (20 + 40 + 10) = 70, and 40/70 = 0.5714..., which is about 57%.
The correct choice is Choice (4).
15. An equation that yields the same solutions as x2 − 10x − 24 = 0 is
(1) (x + 5)2 = 1
(2) (x - 5)2 = 1
(3) (x + 5)2 = 49
(4) (x - 5)2 = 49
Answer: (4) (x - 5)2 = 49
Start by Completing the Square.
x2 − 10x − 24 = 0
x2 − 10x + 25 − 24 = 25
x2 − 10x + 25 = 49
(x - 5)2 = 49
The correct choice is Choice (4).
16. The function f(x) is shifted three units right and four units up. The result of this transformation is
(1) f(x − 3) + 4
(2) f(x + 3) + 4
(3) f(x + 4) - 3
(4) f(x - 4) - 3
Answer: (1) f(x − 3) + 4
If the function is shifted up four units, then four will be added to the function OUTSIDE of the parentheses. Eliminate Choices (3) and (4).
When a function is shifted to the right three units, then three will be subtracted INSIDE the parentheses.
The correct choice is Choice (1).
17. Hana was asked to solve a quadratic equation. Her first step is shown below.
Step 1: x2 − 8x + 16 = 3 + 16
The property that Hana used is the
(1) distributive property
(2) commutative property
(3) additive inverse property
(4) addition property of equality
Answer: (4) addition property of equality
The same amount is added to each side of the equal sign. If x = y then x + n = y + n.
The correct choice is Choice (4).
18. A data set is given below:
What is the value of the upper quartile of this data set?
(1) 28
(2) 32
(3) 34
(4) 42
Answer: (3) 34
There are 10 pieces of data. The upper quartile is the median of the upper half of the data (the top five numbers). That number is 34.
The upper quartile (Q3) is 34, which is Choice (3).
19. When −3x2 + 7x − 1 is subtracted from 2x2 − 3x + 10, the result is
(1) 5x2 + 4x + 9
(2) 5x2 - 10x + 11
(3) -5x2 + 4x + 9
(4) -5x2 + 10x - 11
Answer: (2) 5x2 - 10x + 11
When subtracting polynomials, what follows the "from" goes on top:
2x2 − 3x + 10
−3x2 + 7x − 1
5x2 - 10x + 11
The correct choice is Choice (2).
20. The value of a home in Buffalo can be modeled by the function V(t) = 96,949(1.0448)t , where V(t) is the value of the house after t years. What is the percent of increase in the value of the home each year?
(1) 1.0448%
(2) 0.0448%
(3) 0.448%
(4) 4.48%
Answer: (4) 4.48%
The 1.0448 means that the value is increasing by a factor of 0.0448, which is 4.48%
The correct choice is Choice (4).
21. A rod is an old English measure of distance that is equivalent to 5.5 yards. How many inches are 2.5 rods? [1 yard = 3 feet]
(1) 66
(2) 165
(3) 198
(4) 495
Answer: (4) -51.9
There are 5.5 yards in one rod. There are 3 feet in one yard. There are 12 inches in one foot.
Therefore, there are 2.5 * 5.5 * 3 * 12 = 495 inches in 2.5 rods. This is Choice (4).
Choice (1) is the answer you would get if you multiply 5.5 * 12 and forgot the 3 and the 2.5.
Choice (2) is the answer you would get if you forgot to multiply by 3.
Choice (3) is the number of inches in one rod.
I'm surprised none of the incorrect choices had the number of feet, which is 41.25.
22. When solving the equation 2x2 − 3x − 6 = 0 using the quadratic formula, the solutions are
(1) (3 + √(57)) / 4
(2) (3 + √(39)) / 4
(3) (-3 + √(57)) / 4
(4) (-3 + √(39)) / 4
Answer: (1) (3 + √(57)) / 4
Using the quadratic formula in this case, a = 2, b = -3, and c = -6. This means that (-b) = (-(-3)) = 3. Eliminate Choices (3) and (4).
The discriminat has the formula b2 - 4ac. Remember to use parentheses!
Thus, (-3)2 - 4(2)(-6) = 9 - (-48) = 57. Eliminate Choices (2) and (4).
The correct choice is Choice (1).
23. For which function is the axis of symmetry x = −4?
(1) f(x) = -x2 - 4x - 1
(2) g(x) = -x2 + 8x + 5
(3) h(x) = x2 + 8x + 3
(4) k(x) = x2 + x - 4
Answer: (3) (x - 3)2 = 33
You can graph each of these parabolas and check. Or you can use the formula x = -b/(2a) to see which is x = -4.
In Choice (1), x = -(-4) / (2(-1) = -2. Eliminate Choice (1).
In Choice (2), x = -(8) / (2(-1) = 4. Eliminate Choice (2).
In Choice (3), x = -(8) / (2(1) = 4. This is the correct answer.
In Choice (4), x = -(1) / (2(1) = -1/2. Eliminate Choice (4).
24.In simplest radical form, the product of 2√(6) and 5√(3) is
(1) 10
(2) 21
(3) 30√(2)
(4) 10√(18)
Answer: (3) 30√(2)
Multiply the terms and then simplify the radical.
(2√(6))(5√(3))
= (10)√( (6)(3) )
= (10)√( (2)(3)(3) )
= (3)(10)√(2)
= 30√(2)
This is Choice (3).
Note that Choice (4) is the same amount, but it isn't simplified. The radicand 18 can be factored into (9)(2) and (9) is a perfect square.
More to come. Comments and questions welcome.
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