If 2√x = x – 3, which of the following is the solution set for x ?

Solution
Plug In the Answers! The answers aren't in order, and some numbers appear more than once, so you don't need to start in the middle.
Instead, start with 9 because it is in three of the four choices.
If x = 9, then 2√9 = 9  3.
√9 = 3, so the left side of the equation is 2 × 3 = 6, and the right side of the equation is 9  3 = 6.
This works, so 9 is part of the solution set; eliminate (B) because it doesn't include 9.
Next, try x = 1: 2√1 = 1  3, which solves to 2 = 2.
This isn't true, so 1 is not part of the solution set; eliminate (D).
Lastly, try x = 1: 2√1 = 1  3.
You cannot take the square root of a negative number, so this doesn't work.
Eliminate (A) and choose (C).
What is the value of (2 + 8i)(1 – 4i) – (3 – 2i)(6 + 4i) ?
(Note: i = √1 )

Solution
Taking note that i = √1, the expression (2 + 8i)(1  4i)  (3  2i)(6 + 4i) becomes \(\left(2 + 8\sqrt{1} \right) \left(1  4\sqrt{1} \right)\)  \(\left(3  2\sqrt{1} \right) \left(6 + 4\sqrt{1} \right)\).
Expanding, this becomes \(2  8\sqrt{1} + 8\sqrt{1}  32\left (\sqrt{1} \right )^{2}\)  \(\left(18 + 12\sqrt{1}  12\sqrt{1}  8\left (\sqrt{1} \right )^{2} \right)\)
\(= 2  32\left (\sqrt{1} \right )^{2}  18 + 8\left (\sqrt{1} \right )^{2}\)
\(= 8\left (\sqrt{1} \right )^{2}  32\left (\sqrt{1} \right )^{2}  16\).
This further simplifies to 8 + 32  16 = 8. This is (A).
Cuthbert is conducting a chemistry experiment that calls for a number of chemicals to be mixed in various quantities. The one amount of which he is unsure is grams of potassium, p . If Cuthbert is certain that (3p^{2} + 14p + 24) – 2(p^{2} + 7p + 20) = 0, what is one possible value of 3p + 6, the exact number of grams of potassium that Cuthbert would like to use for this experiment?

Solution
Begin by simplifying the equation given. (3p^{2} + 14p + 24)  2(p^{2} + 7p + 20) = 3p^{2} + 14p + 24  2p^{2}  14p  40 = p^{2}  16 = 0.
Factoring the left side of the simplified equation, we find that (p  4)(p + 4) = 16.
Solving for p, we find that p = ±4.
The value of 3p + 6 must then be either 3(4) + 6 = 6 or 3(4) + 6 = 18.
The latter value is (B).
Samantha offers two different packages of yoga classes at her yoga studio. She offers two hot yoga sessions and three zero gravity yoga sessions at a total cost of $400. She also offers four hot yoga sessions and two zero gravity sessions at a price of $440. Samantha wants to offer a larger package for longtime clients in which the cost must exceed $800. If Samantha does not wish to include more than 13 sessions for the longtime client package, will she be able to create this package for her clients?

Solution
Translate from English to math in bitesized pieces.
Make the price of a hot yoga lesson h and the price of a zero gravity yoga session z.
If she offers 2 hot yoga and 3 zero gravity yoga sessions for $400, then 2h + 3z = 400.
Similarly, if 4 hot yoga and 2 zero gravity yoga sessions are $440, then 4h + 2z = 440.
Now, be sure to Read the Full Question: You want to know whether Samantha can create a package that's greater than $800 but has fewer than 13 sessions.
If you stack the two equations and then add them together, you get 6h + 5z = 880.
In other words, she can offer 6 hot yoga and 5 zero gravity yoga sessions (11 total sessions) for $880.
This satisfies her requirements, so you know the answer is "Yes"; eliminate (A) and (B).
For (C), because you don't know the price of each lesson individually, you don't know yet whether 5 hot yoga and 5 zero gravity yoga sessions will be over $800; leave (C) for now.
For (D), if 6 hot yoga and 5 zero gravity yoga sessions were over $800,then adding a zero gravity yoga session will still be over $800. Given what you already know, (D) must be true; choose (D).
If x^{2} + 2xy + y^{2} = 64 and y – x = 12, which of the following could be the value of x ?

Solution
Factoring the left side of the equation x^{2} + 2xy + y^{2} = 64 gives (x + y)^{2} = 64.
Taking the square root of both sides of the equation, we find that x + y = 8 or 8.
The other equation provides that y  x = 12, so y = x + 12 .
Substitute this value of y into the first equation: either x + (x + 12) = 8, so 2x + 12 = 8, 2x = 4, and x = 2, or else or x + (x + 12) = 8, so 2x + 12 = 8, so 2x = 20, and x = 10.
Therefore, x could be either 2 or 10, and only 10 is an option in the answers, so (A) is correct.
2c + 3d = 17
6c + 5d = 39
In the system of linear equations above, what is the value of 4c – 4d ?

Solution
We must find values of c and d by solving the system of equations in order to determine the value of 4c  4d.
There are several ways to go about this. One way is to multiply the terms of the equation 2c + 3d = 17 by 3 to get 6c  9d = 51 .
If you stack and add this equation with the second equation, the result is 4d = 12, which solves to d = 3.
Plug this value for d into the equation 6c + 5d = 39 to get 6c + 15 = 39, so 6c = 24 and c = 4.
Therefore, 4c  4d = 4(4)  4(3) = 16  12 = 4. This is (C).
Steven needs to buy t theme park tickets for himself and his family. Each ticket costs $80, and the number of tickets he needs to buy can be modeled by the expression t^{2} – 4t – 90 = 6 when t > 0. What is the total cost of the theme park tickets that Steven purchased?

Solution
Rearranging and factoring the expression provided in the question, we have t^{2}  4t  90 = 6 → t^{2}  4t  96 = 0 → (t  12)(t + 8) = 0.
Therefore, t  12 = 0 and t + 8 = 0. t must then equal 12 or 8.
If t represents the number of tickets Steven buys, then only t = 12 is consistent with the context of the question.
If each ticket costs $80, Steven must have spent $80 × 12 = $960.
This is (C).
Given the equation y = 3x^{2} + 4, what is the function of the coefficient of 3 ?

Solution
When a function f(x) is transformed into a function of the form f(ax), where a is a constant, if a > 0, the function will be compressed horizontally by a factor of a.
Here, y = x^{2} + 4 can be represented as the parent function, and y = 3x^{2} + 4 as the transformed function compressed horizontally versus the parent function, and thus narrower, by a factor of 3. This is (D).
If you're not sure, try plugging values into each equation to construct a rough graph of each equation and compare them.
If 3a + 2b = 24 and 4a + 5b = 53, what is the value of a + b ?

Solution
This question requires evaluating both equations to determine the values of a and b.
You could begin by solving either of the two equations for a or b, and then substituting the solution into the other equation.
But note that the question asks for the value of a + b, so check to see if there's a faster way: Could you stack and add (or subtract) the equations?
If you stack and add the equations, you get 7a + 7b = 77.
Now divide both sides of the equation by 7, resulting in a + b = 11.
This is (D).
Ryan and Allison build a ramp to help their elderly cat, Simms, walk up to their bed. They need the ramp to make a 35° angle with their bedroom floor. How long must the ramp be to reach the top of their bed that is exactly three feet off the ground?

Solution
The question describes a ramp that forms a triangle, the length of which is the hypotenuse of the triangle.
The height of the ramp (3 feet) is the length of the side of the triangle opposite the 35° angle.
In general for some angle θ, sin θ = \(\frac{opposite}{hypotenus}\).
In the question, this corresponds to sin 35° = \(\frac{opposite}{hypotenus} = \frac{3}{length \; of \; ramp}\).
∴ length of ramp = \(\frac{3}{sin35^{\circ}}\).
This is (D).