Algebra Problems of the Day (Algebra 1 Regents, June 2023 Part I)

The following questions appeared on the une 2023 Algebra 1 Regents Exam

More Regents problems.

Algebra 1 Regents, June 2023

Part I: Each correct answer will receive 2 credits.


1. The expression 9m² - 100 is equivalent to

1) (3m - 10)(3m + 10)
2) (3m - 10)(3m - 10)
3) (3m - 50)(3m + 50)
4) (3m - 50)(3m + 50)

Answer: 1) (3m - 10)(3m + 10)

The given expression is a difference of two perfect squares, which factors into two conjugates, two expressions with the same terms, but one has a plus and the other has a minus.

If you look at the choices, only one has a pair of constants that will have a product of -100 when multiplied, and that is Choice (1).

Choice (2) has a product of 9m² - 60m + 100.
Choice (3) has a product of 9m² - 2500.
Choice (4) has a product of 9m² - 300m + 2500.

2. Which expression represents an irrational number?

1) √(16) + √(1)
2) √(25) + √(4)
3) √(36) + √(7)
4) √(49) + √(9)

Answer: 3) √(36) + √(7)

The sum of two rational numbers is always rational. The sum of a rational and in irrational number is always irrational.

If you take the square root of a perfect square, you will get a rational number. Out of all the numbers which appear in the choices above, only √(7) is irrational. So the correct answer is Choice (3).

3. Which linear equation represents a line that passes through the point (-3,-8)?

1) y = 2x - 2
2) y = 2x - 8
3) y = 2x + 13
4) y = 2x - 14

Answer: 1) y = 2x - 2

You could put all four equations into your graphing calculator and check the Table of Values for the correct equation for x = -3. Or you can just substitute x = -3 into each of this equations. Since they all have the same slope, at most one of the equations will work.

y = 2(-3) - 2 = -8. This is correct, so the answer is Choice (1).
y = 2(-3) - 8 = -14, so (-3,-8) is not on this line.
y = 2(-3) + 13 = 7, so (-3,-8) is not on this line.
y = 2(-3) - 14 = -20, so (-3,-8) is not on this line.

4. The expression (5x² - x + 4) - 3(x² - x - 2) is equivalent to

1) 2x² - 2x + 2
2) 2x² + 2x + 10
3) 2x⁴ - 2x² + 2
4) 2x⁴ - 2x² + 10

Answer: 2) 2x² + 2x + 10

When subtracting polynomials, the exponents DO NOT change. Eliminate Choices (3) and (4) immediately.

When you use the Distributive Property on the second expression, make sure you carry the minus sign. You are multiplying by NEGATIVE three.

(5x² - x + 4) - 3(x² - x - 2)
5x² - x + 4 - 3x² + 3x + 6
2x² + 2x + 10

Choice (2) is the correct answer.

5. 5 The 24th term of the sequence 25, 211, 217, 223, … is

1) -149
2) -143
3) 133
4) 139

Answer: 2) -143

Each given term is negative and the value is decreasing. So the 24th term will be a negative number. Eliminate Choices (3) and (4).

Each term in the sequence is 6 less than the one before it. The common difference is -6.

If the first term is -5, and the common difference is -6, then the 24th term will be -5 - 6(23), because you have to subtract -6 twenty-three times to get to the 24th term.

-5 - 6(23) = -143, which is Choice (2).

If you made a mistake and multiplied by 24, you would've gotten -149, which is Choice (1), which would be an incorrect response.

6. When completing the square for x² - 18x + 77 = 0, which equation is a correct step in this process?

1) (x - 9)² = 4
2) (x - 3)² = 2
3) x = +13
4) (x - 9) = +9

Answer: 1) (x - 9)² = 4

When you Complete the Square, you need half of the coefficient of the middle term (when the x² term has a coefficient of 1). So (x - 3)² is incorrect, and can be eliminated.

To Complete the Square, do the following:

x² - 18x + 77 = 0
x² - 18x = -77
(half of -18 is -9, and (-9)² = +81)

x² - 18x + 81 = -77 + 81
x² - 18x + 81 = 4
(x - 9)² = 4

This is Choice (1).

If you expand (x - 9)² = 4, you get
x² - 18x + 81 = 4, which when you subtract 4 from both sides becomes,
x² - 18x + 77 = 0.

7. Which function will have the greatest value when x > 1?

1) g(x) = 2(5)^x
2) f(x) = 2x + 5
3) h(x) = 2x² + 5
4) k(x) = 2x³ + 5

Answer: 1) g(x) = 2(5)^x

You could put all four equations into your graphing calculator and look at the results on the screen and in the table of values.

It should be obvious that quadratic functions will rise faster than linear functions when x > 1. Also, cubic functions will rise faster than quadratic functions when x > 1. Finally, exponential functions will rise faster than cubic functions when x > 1.

The correct answer is Choice (1).

When x = 1, f(1) = h(1) = k(1) = 7, and g(1) = 10.

8. Mike uses the equation b = 1300(2.65)^x to determine the growth of bacteria in a laboratory setting. The exponent represents

1) the total number of bacteria currently present
2) the percent at which the bacteria are growing
3) the initial amount of bacteria
4) the number of time periods

Answer: 4) the number of time periods

The exponent is how many times you are going to multiply by 2.65.

The total number of bacteria is b.

The percent at which the bacteria are growing is 165%. (That is, 1 + 1.65 = 2.65.)

The initial amount of bacteria is 1300.

More to come. Comments and questions welcome.

More Regents problems.

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