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The solution is (z,w) = (5 + 7i, 2 - 4i).

The answer is 335. 

We have

\(33 \cdot 781 = 27335\)

So 335 is our answer. 

Thanks! :)

First, let's cross multiply. We have

\(\sqrt{x}\left(2x\sqrt{6}+4\right)=\left(x\sqrt{3}+\sqrt{2}\right)\cdot \:1\)

Now, we do some simplifying. Factoring and expanding out everything, we have

\(2\sqrt{x}\left(\sqrt{6}x+2\right)=\sqrt{3}x+\sqrt{2}\)

Squaring both sides of the equation, we find that

\(24x^3+16\sqrt{6}x^2+16x=3x^2+2\sqrt{6}x+2\)

Solving for this, we get

\(x=\frac{1}{8}\)

So x is 1/8.

So our answer is 1/8. 

Thanks! :)

There are 487 ways to choose the 3 circles. 

We can write an equation to solve this problem. 

First, let's note that every number in the form 5 modulo 171 can be written in the expression

\(171x+5\) where x is an integer. 

We set this to equal 1000 to get the largest number possible. we have

\(171x+5 = 1000\)

We are trying to find x. We have

\(171x=995\)

Dividing, we get

\(x=5.81871345029\)

We actually have to round down, so we get x is 5. 

So our final answeer is 5. 

Thanks! :)

Let's make some observations. 

2 modulo 7 means that it leaves a remainder of 2 when divided by 7. 

Well, let's look at the numbers

\(1, 2, 3, 4, 5, 6, 7 \)

Note that other than 2, other numbers leave a remainder of NOT 2. 

So our answer is just \(2\)

Thanks! :)

Now, let's note something really important. 

To determine the units digit of a number, we must focus on the unit digits of the two numbers we are multiplying. 

Any other term actually doesn't matter. 

Thus, we have

\(3*7=21\)

So the units digit is 1. 

Also, we could multiply these two numbers together, since both are small. 

\(13*287=3731\)

The units digit is 1. 

So our final answer is 1. 

Thanks! :)

Fill in the blanks to make a quadratic whose roots are $1$ and $-1.$   
x^2 + ___ x + ___   

Since the roots are +1 and –1, the factors are (x – 1) and (x + 1)    

Multiplied together gives x² – 1   

So, the first space contains a     0    

the second space contains a    –1   

_.

Find the largest value of $x$ such that $3x^2 + 17x + 15 = 2x^2 + 21x + 12 - 5x^2 + 17x + 34.$   

rearrange terms for clarity              (3x² – 2x² + 5x²) + (17x – 21x – 17x) + (15 – 12 – 34)  =  0   

combine like terms                          6x² – 21x – 31  =  0   

A parabola that opens upward has no largest value.   

_.

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)

Ok, we can go one by one to solve these problems. 

Hoever, let's note that if we have f(x) = x, then we set the right side of the function to equal x.

Then, we see if the x value satisfies the conditions given. 

First, we have \(f(x) = 2x + 8\). Setting this to equal x, we fuind that \(x=2x+8\)

Solving for x, we find that \(x=-8\). This doesn't satisfy the condition \(1 \le x \le 2\), so it is not a valid solution. 

Next, we have \(f(x) = 13 - 5x\). Setting this to equal x, we have \(x=13-5x\)

Finding the value of x, we get \(x=13/6\). This satisfies the inerval, so it is a solution. 

Third, we have \(f(x) = 20 - 14x\). Since this equals x, we can write \(x = 20 - 14x\) 

Trying to find x, we have that \(x=20/15=4/3\). This does NOT satisfy the interval, so it is not a solution. 

Lastly, we have \(f(x) = 40 + 5x \). setting this function to equal x, we have \(x=40+5x\)

Solving for x, we get \(x=-10\), which is invalid. 

Thus, the only value that works i 13/6. 

So our answer is 13/6. 

Thanks! :)

In order to solve this problem, let's analyze somethings about it. 

First, let's note that right triangles \(ABC, ABE, ABD, BHD, BEC\) are all 30-60-90 triangles. 

Also, note that right triangles \( ABE,ADC \) are 45-45-90 triangles. 

In order to solve the problem, let's draw a line to help find HCA for us. 

Let's draw the altitude of ABC by connecting c to line segment AB. 

This creates a right triangle!

Since we know that

\(\angle BAC =60^\circ\)

HCA would have to be 30 degrees because of the right triangle.

So we have

\(\angle HCA = 30^\circ\)

So 30 degrees is our answer. 

Thanks! :)

To determine the value of k for which the system has no solution, we can first rewrite the system in matrix form:

| 1 1 3 | | x | | 10 |
| -4 2 5 | * | y | = | 7 |
| k 0 1 | | z | | 3 |

To check if the system has no solution, we need to calculate the determinant of the coefficient matrix and the determinant of the augmented matrix. If both determinants are nonzero and not equal to each other, then the system has no solution.

The determinant of the coefficient matrix is:
det = 1(2*1 - 0*5) - 1(-4*1 - 3*5) + 3(-4*0 - 2*5) = 2 + 17 - 30 = -11

The determinant of the augmented matrix is:
det_aug = 1(2*3 - 0*7) - 1(-4*3 - 5*7) + 3(-4*0 - 2*7) = 6 + 29 - 0 = 35

Since the determinants are not equal (det ≠ det_aug) and det is not equal to zero, there is no solution for the system.

Therefore, for the system to have no solution, the value of k must be such that the determinant of the coefficient matrix is equal to zero:

-11 + k(0+4) = 0

-11 + 4k = 0

4k = 11

k = 11/4

Therefore, the value of k for which the system has no solution is k = 11/4.

The expression under the logarithm cannot be less than zero because the logarithm of a negative number is undefined.

Therefore, -20x + 12sqrt(x) must be greater than zero. This means:

12sqrt(x) > 20x

Square both sides of the inequality:

144x > 400x^2 - 480x + 144

Rearrange the inequality to get a quadratic expression:

400x^2 - 624x + 144 = 0

Divide both sides by 48 to simplify:

25x^2 - 39x + 9 = 0

This expression factors as:

(5x - 3)(5x - 3) = 0

Therefore, the possible solutions for x are:

x = 3/5

However, the original expression includes a square root. Since the square root of a negative number is undefined, we need to check if x = 3/5 makes -20x + 12sqrt(x) positive.

Plugging in x = 3/5:

-20(3/5) + 12sqrt(3/5) = -12 + 12sqrt(3/5)

Since 12sqrt(3/5) is positive, -12 + 12sqrt(3/5) is also positive. Therefore, x = 3/5 is a valid solution.

Answer: x = 3/5

We can solve this equation by utilizing the Pythagorean identity, which states that cos^2(x) + sin^2(x) = 1.

Here's how we proceed:

Rewrite the equation: We can rewrite the given equation as cos^2(x) - sin^2(x) - sin(x) = 0.

Factor the equation: Noticing a common factor of cos^2(x) - 1, we can factor the equation as (cos^2(x) - 1) * (sin(x) + 1) = 0.

Separate the factors: This gives us two possibilities:

cos^2(x) - 1 = 0

sin(x) + 1 = 0

Solve for cos(x):

From cos^2(x) - 1 = 0, add 1 to both sides and take the square root of both sides: cos(x) = ±1. Since 0 < x <= 2*pi, the solutions for cos(x) are x = 0 and x = pi.

Solve for sin(x):

From sin(x) + 1 = 0, subtract 1 from both sides: sin(x) = -1. However, within the range 0 < x <= 2*pi, there is no solution for sin(x) to be -1.

Combine solutions: Therefore, the solutions for the equation within the given range are x = 0 and x = pi.

Answer: The solutions for the equation cos^2(x) - sin^2(x) = sin(x) within the range 0 < x <= 2*pi are x = 0 and x = pi.

In order to know what the 43rd term is, we must know the common difference. 

Luckily, the problem basically tells us what the difference is. 

Let's let that difference be d. We have from the problem that

\(1/4 + 10d = -1/5 \\ 10d = -1/5 - 1/4 = -9/20 \\\ d = -9/200\)

Now, we biuld off of the 33rd term. Since we know it is 1/5, we simply just have to add the d a number of times. 

We have

\( -1/5 + 10d = -1/5 + 10 (-9/200) = -1/5 - 9/20 = -65/100 = -13 / 20\)

So our answer is -13/20

Thanks! :)

Let's first focus on the first equation. 

Squaring both sides of the first equation, we find that

\(a^2 + 2ab + b^2 = 16 \)

Reaaranging the second equation from the problem, we get that

\(a^2 -ab+ b^2 = 6\)

Now, moving on, let's note something important. We have that

\(a^3 + b^3 = (a + b) ( a^2 + b^2 - ab) \)

We already have all the terms needed to solve the problem. We know every single number. 

The first term parenthesis is the first equation, and the second parenthesis is the second equation. 

Thus, plugging in 6 and 4, we get that

\(a^3 + b^3 = 6*4 =24\)

So 24 is our answer. 

Thanks! :)

Let's note something for this problem first. 

We are basically taking a look at the squares of numbers, and eeing the largest 3 digit number. 

Thus, let's test some integers. We find that

\(31^2 = 961 \\ 32^2 = 1024 \)

This shows that 31 is the largest value squared. 

Thus, 961 is our answer. 

So we have \(x=961\)

Thanks! :)

Let's first multiply the terms together and see what we get. 

Multiplying \(x^3(x^2+tx)\), we use the distrbutive property. 

Factoring in x^3, we get that

\(x^5+tx^4\)

There is no x^2 term in this expansion of the product. 

This means t can pretty much be anything unless it contain x^-2. 

Other than that, t could be pretty much any number or polynomial. 

Thanks! :)

Now, there are two ways to do this problem. 

First off, let's look at the possibilities of how we can create x. 

We have x and 14, which gets us 14x. 

Next, we have -23x and 1, which gives us -23x. 

Thus, the coefficient is simply \(14x-23x = -9x\)

We could also just expand the entire thing. We find that after expanding out the entire thing, we have

\(x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14\)

Thus, the coefficient of x is just \(-9\)

So -9 is our answer. 

Thanks! :)