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).

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).

Begin by simplifying the equation given. (3p² + 14p + 24) - 2(p² + 7p + 20) = 3p² + 14p + 24 - 2p² - 14p - 40 = p² - 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).

Translate from English to math in bite-sized 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).

Factoring the left side of the equation x² + 2xy + y² = 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.

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).

Rearranging and factoring the expression provided in the question, we have t² - 4t - 90 = 6 → t² - 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).

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 = x² + 4 can be represented as the parent function, and y = 3x² + 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.

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).

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).