Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, June 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2011

Part I: Each correct answer will receive 2 credits.


21. The solution set of √(3x + 16) = x + 2 is

1) {-3, 4}
2) {-4, 3}
3) {3}
4) {-4}

Answer: 3) {3}

You can solve this or you can check the answers. I would start by checking x = 3 because if it is or is not (either way), you eliminate two choices.

√(3(3) + 16) = (3) + 2

√(25) = 5

5 = 5

We know x = 3 is a solution, so eliminate Choices (1) and (4). Check x = -4.

√(3(-4) + 16) = (-4) + 2

√(4) = -2

The square root function cannot return a negative answer. So this is eliminated.

Working it out algebraically:

√(3x + 16) = x + 2

3x + 16 = x² + 4x + 4

x² + x - 12 = 0

(x + 4)(x - 3) = 0

x = -4 or x = 3

However, you have to check if you picked up an extra solution when we squared the radical. We did that above when we discovered that x = -4 is not a valid solution.

22. Brian correctly used a method of completing the square to solve the equation x² + 7x − 11 = 0. Brian’s first step was to rewrite the equation as x² + 7x = 11. He then added a number to both sides of the equation. Which number did he add?

1) 7/2
2) 49/4
3) 49/2
4) 49

Answer: 2) 49/4

To complete the square, divide the middle number by 2 (actually, by 2a, but in this case a = 1), and then square that quotient.

7 divided by 2 is 7/2, which when squared becomes 49/4, which is Choice (2).

23. The expression (sin² θ + cos² θ) / (1 - sin² θ) is equivalent to

1) cos² θ
2) sin² θ
3) sec² θ
4) csc² θ

Answer: 3) sec² θ

The numerator has a sum equal to 1, which is the radius of the unit circle.

If you replace the 1 in the denominator with sin² θ + cos² θ, you will be left with just cos² θ.

1 / cos² θ = sec² θ

24. The number of minutes students took to complete a quiz is summarized in the table below.

If the mean number of minutes was 17, which equation could be used to calculate the value of x?

1) 17 = (119 + x) / x
2) 17 = (119 + 16x) / x
3) 17 = (446 + x) / (26 + x)
4) 17 = (446 + 16x) / (26 + x)

Answer: 4) 17 = (446 + 16x) / (26 + x)

To find the mean, you have to divide the total number of minutes by the total numbers of students.

The number of students is 26 + x. Eliminate Choices (1) and (2).

The total number of minutes is the sum of the products of students times minutes in each column. This is because the actual data is 14, 14, 14, 14, 14, 15, 15, 15, etc.

That means that you want 446 + 16x not 446 + x, because x students took 16 minutes.

More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon. Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides. Available in softcover or ebook at Amazon. If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.