Algebra Problems of the Day (Integrated Algebra Regents, June 2013)

While I'm waiting for new Regents exams to come along, I revisiting some of the older NY Regents exams.

More Regents problems.

Administered June 2013

Part I: Each correct answer will receive 2 credits.


26. If the roots of a quadratic equation are -2 and 3, the equation can be written as

(1) (x - 2)(x + 3) = 0
(2) (x + 2)(x - 3) = 0
(3) (x + 2)(x + 3) = 0
(4) (x - 2)(x - 3) = 0

Answer: (2) (x + 2)(x - 3) = 0

Since the signs of the two solutions are different, you can eliminate the two choices where the signs are the same. That is, Choices (3) and (4) are gone.

If 3 is a root, then it is a "zero" of the equation and (x - 3) must equal 0. Likewise, -2 implies that (x - (-2)) or (x + 2) = 0

27. Which equation represents a line that is parallel to the y-axis and passes through the point (4,3)?

(1) x = 3
(2) x = 4
(3) y = 3
(4) y = 4

Answer: (2) x = 4

Parallel to the y-axis means that it is a vertical line. Vertical lines have equations such as x = a. The x-coordinate is 4, so x = 4 is the equation of the line.



28. There are 18 students in a class. Each day, the teacher randomly selects three students to assist in a game: a leader, a recorder, and a timekeeper. In how many possible ways can the jobs be assigned?
(1) 306
(2) 816
(3) 4896
(4) 5832

Answer: (3) 4896

There are 18 * 17 * 16 ways to select the three positions. Order matters, so this is a permutation problem, not a combination problem.

THere are 18 ways to pick the first assistant, 17 choices for the second, and 16 for the third. The Counting Principle says to multiple all three of them.

29. In triangle RST, angle R is a right angle. If TR = 6 and TS = 8, what is the length of RS?

(1) 10
(2) 2
(3) 2 SQRT(7)
(4) 7 SQRT(2)

Answer: (3) 2 SQRT(7)

If R is the right angle, the line TS, which is 8, is the hypoetenuse, and RS, which must be shorter than 8, is another leg.

This is NOT a 6-8-10 right triangle.

6² + RS² = 8²
36 + RS² = 64
RS² = 28
RS = SQRT(28) = SQRT(4)*SQRT(7)
RS = 2 SQRT(7)

If you didn't know how to simplify SQRT(28), you could have gotten a decimal from your calculator, and then checked it against Choices (3) and (4) in your calculator.

Note that 10 is longer than the hypotenuse, but a likely mistake to make. Choice (2) can be eliminated because 2, 6, 8 does NOT make a triangle. It makes a striaght line.

30. How many solutions are there for the following system of equations? y = x² - 5x + 3
y = x - 6

(1) 1
(2) 2
(3) 3
(4) 0

Answer: (1) 1

First of all, you cannot have 3 solutions to a quadratic-linear system of equations. The line can intersect the curve at two points, at one point, or not intersect at all.

You can put the two equations into you graphing calculator and look to see the answer.

Or you can solve the equation. Again, you don't need to know the solutions, just how many there are.

x² - 5x + 3 = x - 6
x² - 6x + 9 = 0
(x - 3)(x - 3) = 0
x - 3 = 0
x = 3
This is one solution.

This was a simple quadratic to solve. However, had it been more complicated, you might have chosen to check the discriminant, b² - 4ac, to see if it was positive, zero, or negative.

In this case, (-6)² - 4(1)(9) = 36 - 36 = 0, so there is one solution.

End of Part I



More to come. Comments and questions welcome.

More Regents problems.