ABJeIIy

The answer is f(17) = -21.

For this problem, you must consider magnitude and argument of complex numbers geometrically.

The answers are as follows:

r_1 = 16

theta_1 = 3*pi/2

r_2 = sqrt(3)

theta_2 = 5*pi/4

r_3 = 12

theta_3 = pi/3

For this problem, you must consider magnitude and argument of complex numbers geometrically.

The answers are as follows:

r_1 = 16

theta_1 = 3*pi/2

r_2 = sqrt(3)

theta_2 = 5*pi/4

r_3 = 12

theta_3 = pi/3

For this problem, you must consider magnitude of complex numbers geometrically.

For the first part, the answers are |v| = 2, |w| = 3, |z| = 3.

For the second part, the answers are |v + w| = sqrt(2), |w + z| = 2, |z + v| = sqrt(3).

For this problem, you must consider magnitude of complex numbers geometrically.

For the first part, the answers are |v| = 2, |w| = 3, |z| = 3.

For the second part, the answers are |v + w| = sqrt(2), |w + z| = 2, |z + v| = sqrt(3).

For this problem, you must consider addition and subtraction of complex numbers geometrically.  The answer turns out to be s, t, u.

For this problem, you must consider addition and subtraction of complex numbers geometrically.  The answer turns out to be s, t, u.

The coordinates of the point are (2/3*(-6) + 1/3*(15), 2/3*19 + 1/3*(-5)) = (1,11).

Let's break down the expression step by step.

sqrt(49 - 20*sqrt(6)) = ?

First, we can start by evaluating the expression inside the parentheses:

49 - 20*sqrt(6) = ?

We can use the distributive property to multiply 20 by sqrt(6):

49 - 20*sqrt(6) = 49 - 20(sqrt(6))

Now, we can simplify this expression by recognizing that 49 is a perfect square:

49 = 7^2

So, we can rewrite the expression as:

49 - 20(sqrt(6)) = 7^2 - 20(sqrt(6))

Next, we can use the difference of squares formula to simplify further:

a^2 - b^2 = (a + b)(a - b)

In this case, a = 7 and b = sqrt(6), so we get:

7^2 - sqrt(6)^2 = (7 + sqrt(6))(7 - sqrt(6))

Now, we can simplify the expression by combining like terms:

(7 + sqrt(6))(7 - sqrt(6)) = ?

To combine these terms, we can use the fact that sqrt(x)^2 = x:

7 + sqrt(6) and 7 - sqrt(6) are both factors of (7)^2 - (sqrt(6))^2

So, we can factor out a common factor of (7 - sqrt(6)):

= (7 - sqrt(6))(7 + sqrt(6))

And that's the simplified form of the original expression!

sqrt(49 - 20*sqrt(6)) = 7 - sqrt(6)

Now, we want to find the ratio PQ/FH. We can find the lengths of PQ and FH by using similar triangles. Let's look at triangle HFG and triangle GNF.

Since HM is parallel to GN, we have angle NFG = angle GFM. Additionally, since FM/MG = 1/2, we have FG = 2GF. Similarly, using triangle HEN and triangle EMF, we find that EH = 3HE.

Now we see that FG:GF = EH:HE = 3:2. Therefore, triangles HFG and GNF are similar.

From here, we know that PQ/FH = QF/FH = HN/HF. By letting HF = a and HN = b, we have a+b = FH. Since HN:NE = 1:3, b/3y = 1/3, b = y. Similarly, we know that a/x = 2x/x = 2, a = 2x.

Now we have PQ/FH = HN/HF = y/(a+b) = y/(2x+y). Let Y = PQ/FH = y/(2x+y). Since y = HN = 1 and x = FM = 1, we have Y = 1/(2+1) = 1/3.

Therefore, the ratio PQ/FH is 1/3.

There are 25,480 ways of coloring the squares.

Problem 2:

We can solve this problem by analyzing the factored expressions and using casework to explore the possible values of x and y.

Factoring the Expressions:

(2x^2 + 4x + 8) can be factored as 2(x^2 + 2x + 4). We can further factor (x^2 + 2x + 4) by noticing it's a perfect square trinomial. So, 2(x^2 + 2x + 4) = 2(x + 2)^2.

(3y^2 - 6y + 7) can be factored as 3(y^2 - 2y + 7/3). This trinomial cannot be factored further with integer coefficients.

Setting Up the Cases: The equation is: 2(x + 2)^2 * 3(y^2 - 2y + 7/3) = 24

We can rewrite it as: (x + 2)^2 * (y^2 - 2y + 7/3) = 4 ---------(1)

Since 4 can be factored as 1 * 4 or 2 * 2, we will consider two cases based on these factorizations:

Case 1: (x + 2)^2 = 1 and (y^2 - 2y + 7/3) = 4

Case 2: (x + 2)^2 = 4 and (y^2 - 2y + 7/3) = 1

Solving Case 1:

(x + 2)^2 = 1 --> x + 2 = ±1 --> x = -1 or x = -3

(y^2 - 2y + 7/3) = 4 --> This quadratic factors as (y - 1)(y - 7/3) = 0. So, y = 1 or y = 7/3.

Therefore, in Case 1, we have four possible ordered pairs: (-1, 1), (-1, 7/3), (-3, 1), and (-3, 7/3).

Solving Case 2:

(x + 2)^2 = 4 --> x + 2 = ±2 --> x = 0 or x = -4

(y^2 - 2y + 7/3) = 1 --> This quadratic has no real number solutions because the discriminant (b^2 - 4ac) is negative.

Therefore, Case 2 has no real number solutions for (x, y).

Total Ordered Pairs: Combining the results from both cases, we have a total of 4 possible ordered pairs from Case 1: (-1, 1), (-1, 7/3), (-3, 1), and (-3, 7/3).

Answer: There are 4 possible ordered pairs (x, y) that satisfy the given equation.

Problem 1:

Let's analyze each statement and see if it's necessarily true given the conditions:

(A) a + c > b + d: Since a > b and c > d, adding them together would maintain this inequality. So, A is true.

(B) 2a + 3c > 2b + 3d: Multiplying both sides of a > b and c > d by positive constants (2 and 3) won't change the direction of the inequality. So, B is true.

(C) a - c > b - d: Subtracting the same number (c) from both sides of a > b doesn't necessarily guarantee a - c > b - d. It depends on the relative values of a, b, and c. So, C is not necessarily true.

(D) ac > bd: Since a > b and c > d, multiplying them together would maintain the inequality if both a and c have the same sign (both positive or both negative). However, if a and c have opposite signs, the product could be negative. So, D is not necessarily true.

(E) a^2 + c^2 > b^2 + d^2: Squaring a positive number increases its value, and squaring a negative number makes it positive. Since a > b and c > d, squaring them preserves the inequality as long as a and c have the same sign. So, E is true.

(F) a^3 + c^3 > b^3 + d^3: Cubing a positive number increases its value significantly, and cubing a negative number makes it much more negative. However, the behavior depends on the signs of a and c relative to b and d. It's not guaranteed that a^3 + c^3 will always be greater than b^3 + d^3. So, F is not necessarily true.

Therefore, the statements that must be true are (A), (B), and (E). The answer is A, B, E.

I found the answer: (a,b,c,d,e) = (4,2,-3,-1,8).