## 1.

The Correct Answer is (C) — This problem is asking you to solve for x. To do that, we just need to get x alone on one side of the equation. We can begin by rewriting the right side of the equation using the distributive property. This give us: 42 = 3x - 12. Next, we can add 12 to both sides. That gives us: 54 = 3x. Now, we just need to divide both sides by 3. That will give us 54/3 = x. 54/3 = 18, so our answer is (C).

## 2.

The Correct Answer is (C) — This problem is asking us to find the value of a variable, k, in a quadratic equation. To accomplish this, all we need to do is rewrite the right side of the equation: (x+3)^2. We can begin by writing out that exponent: (x+3)^2 = (x+3)(x+3). Next, use FOIL to get rid of the parentheses, and group like terms: (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x +9. And there you have it: x^2 + kx + 9 = x^2 + 6x +9, so k = 6, which is answer option (C).

## 3.

The Correct Answer is (C) — This problem is asking you to find the profit for selling one barrel of oil. There are different costs for each step in the process, and some steps occur multiple times, so in order to solve without losing track of anything we can write an algebraic expression. The revenue for the sale, r, is the amount the barrel is sold for, or $93. The costs are:$51 for extraction, e, two transportation steps, t, at $6 each, and three processing steps, p, at$9 each. So we can represent the whole formula for profits as: p = r - e - 2t -3p. We can now plug in the values for each of our variable and solve: p = (93) - (51) - 2(6) - 3(9) = 93 - 51 - 12 - 27 = 42 - 12 - 27 = 30 - 27 = 3, which is answer option (C).

## 4.

The Correct Answer is (B) — This problem asks for the area of the shaded area on the diagram. There’s a long way to solve this, and a short way. First, try drawing a line between Q and R so that you’re dealing with triangles. Next, since we have the area of the square, you could find the dimensions of the square (10x10) by taking the square root of its area (100), then note that if Q is the midpoint of A and C that means that the shaded area can be divided into two right triangles, each of which has one leg, a, of length 10 and another, b, of length 5. Since the area of a right triangle equals (ab)/2, the area of two triangles of equal size is two times (ab)/2 or just ab, which in this case is 50, which is answer option (B).

## 5.

The Correct Answer is (B) — The fastest way to solve this problem is by plugging in some arbitrary value for x (but not 1, 0, or -1) and see what we get. Let’s try plugging in 3. If x = 3, then: 2x/(x - 1) - 3x/(x + 1 ) = 6/2 - 9/4 = (12 - 9)/4 = 3/4. So now, we just need to find the option that’s equal to ¾. (A), (C), and (D) are all negative, so we can rule those all right out. Plugging 3 into (B) shows that it does indeed equal ¾, so we can confirm that it’s the correct answer.

## 6.

The Correct Answer is (A) — This problem is asking you to find an inequality which represents the same range of possible values as a given inequality. The given inequality is an absolute value inequality, but none of our answer options are, so it’s reasonable to start by trying to write the absolute value function out of the equation. The value inside of the absolute value signs may be as great as 5 or as small as -5, so we can rewrite the inequality as -5 <= x-3 <= 5. Unfortunately, that’s not an answer option. However, we can rewrite this inequality to get x alone by adding three to each part of it: -2 <= x <= 8. And, there’s our answer: (A).

## 7.

The Correct Answer is (D) — This problem is asking you to find the value of a, and in order to do that you have to make some inferences with the information that we’re given. There are a few ways to solve this, but a sensible starting point is to write some of the information that we’re given in an algebraic form. We know that: a + b = 132. Since we know that the product of a and b is negative, which implies that one of them is a negative number, but that a is the square of b, which implies that a is a positive number, we know that b is negative. So, b < 0 < a. Because b < 0, a + b < a, so 132 < a. Or, to put that in really plain English: once we know that b is negative, we know that a has to be greater than the sum of a and b. So, the only possible answer option is (D).

## 8.

The Correct Answer is () — This problem describes a function that models the dropping of a stone, and asks you to solve for the number of seconds that it takes for the stone to hit the ground. To solve this problem, we need to take a closer look at the function h(t) = a - t^2. We’ve been told that t = time, and that h(t) = the height of the stone, but we don’t know what a is. We’ve also been told that the stone is dropped from a height of 9 meters, or in other words, that at t = 0 the height of the stone was 9. We can write that in terms of our function as follows: h (0) = a - 0^2 = 9, which tells us that a = 9. Next, we just have to figure out for what t h (t) = 0, or in other words, solve for t in 9 - t^2 = 0. We can add t^2 to both sides to get 9 = t^2, and because 9 is a perfect square we can easily calculate that the square root root of 9 = 3 = t, which gives us our answer: 3.

## 9.

The Correct Answer is () — This problem asks us to determine the length of a line segment, given expressions for the lengths of two other line segments. The first thing is to work out what the relationship of line BC is to AB and AC. BC is a part of line AC: it starts where AB ends, and ends where AC ends. That means that the length of BC will be equal to the difference between the lengths of AB and BC. We don’t need to know what x is in order to work out the difference between x - 4 and x + 6: that’s 10. Since BC is equal to the difference between AB and AC, BC = 10.

## 10.

The Correct Answer is () — The problem is asking us to solve for the value of an expression that includes an imaginary number, which might be a little intimidating if you haven’t worked with imaginary numbers much before. However, the prompt actually gives you all the information that you need about i in order to solve the problem. Just start out doing what you would do if i were any other number, and rewrite this using FOIL: $(1&space;-&space;i\sqrt5)(1&space;+i\sqrt5)=1+i\sqrt5&space;-&space;i\sqrt5&space;-i^{2}\sqrt5&space;=&space;1&space;-i^{2}5$ . Once you take this step, you can see that i has transformed into i^2--which the prompt has already told us is equal to -1. So now, we can just substitute -1 for i in the expression, and solve: 1 - (-1)5 = 1 + 5 = 6.