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Here is one approach:

The value of the first fraction is: \(\boxed{\frac{3}{1}}\)

Find all values of $x$ such that $x^2 - 5x + 4 = -x^2 - 25x - 14$.  

                                                   x² – 5x + 4  =  –x² – 25x – 14   

                                                 2x² + 20x + 18  =  0   

                                                   x² + 10x + 9  =  0   

                                                   (x + 1)(x + 9)  =  0   

                                                             x + 1 = 0   ====>>   x  =  –1   

                                                             x + 9 = 0   ====>>   x  =  –9    

                                                             x  =  –1 , –9   

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When Dirk finishes, Edith has run 14km  and Foley has run 9 km

This means that Foley runs 9/14 as fast as Edith

When Edith has run 24km, Foley has run \(24 (9/14) ≈ 15.43 km\)

So Foley is \(24 - 15.43 ≈ 8.57\) km behind!

Thanks! :)

We can define some variables from the information given from the problem. 

First, since we know that AD bisects BAC, let's name variables

Let  BD  = 3 - x

Let  CD =  x

Thus, we can plug the two numbers in to get the equation

\(BD / AB = CD / AC \\ (3 - x) / 4 = x / 5 \\ 5 (3 - x) = 4x\\ 15 - 5x = 4x \\ 15 = 9x \\ x = 15/9 = 5/3 = CD \)

Finally, plugging in these values for the area, we find that

\([ ADC ] = (1/2) (CD) ( AB) = (1/2) ( 5/3) ( 4) = 10 / 3 \)

Thus, 10/3 is our answer. 

Thanks! :)

Let's graph the question. 

From the graph and the question, we know that

\(CD = BD = 10 \\ ED = ED \\ \angle CDE = \angle BDE = 90\)

So, by SAS congruence, we know that triangle CDE  is  congruent to triangle BDE

Thus, we can complete the problem in a series of steps using this congruence. 

First, we know that

\(sin CED / CD = sin CDE / CE\)

Plugging in some numbers, we find that 

\(sin 75 / 10 = sin 90 / CE \\ CE = 10 sin 75 = BE ≈ 10.35\)

Thus, 10.35 is approximately our answer. 

Thanks! :)

*1800 points

See attached image:

When a triangle rotates around one of it sides, it creates a cone or a bicone.

The formula for the volume of a cone is V = 1/3 h(pi*r^2).

In this situation, the Height is 5, since the cone was rotated around BC.

We know that the triangle is equilateral, so the altitude would be 5/2 * sqrt(3).

This means that the radius is 5/2 * sqrt(3).

V = 1/3 5(pi * (5/2 * sqrt(3))^2)

V = 5/3 (pi * (25/4 * 3))

V = 5/3pi * 75/4

V = 5pi * 25/4

V = \(\frac{125\pi}{4}\)

I will assume you meant THIS: 

 (x^3 + x^2 + x + 1)(x^4 + 3x^3 + 8x^2 + 7x + 1)

When you multiply the two polynomials you will get these parts :

x^2 * 1     +     x * 7x     +     1 * 8x^2       =     16 x^2 

So the coefficient will be ' 16 ' 

The 'x' coefficient  willl come from  the multiplication of the two polynomial factors :   (1 * 8x)  + (7 * x ) = 15x 

so the coefficient of 'x' will be  '15' 

The answer is g(x) = -5x^2 + 7x + 4.

The answer is g(x) = -5x^2 + 7x + 4.

The answer is (sqrt(2) + 1)/3.

The answer is (sqrt(2) + 1)/3.

Here's how we can find the number of 3-digit numbers with exactly 9 factors:

Factorization: A number with 9 factors must be a perfect square of a number with 5 factors (since the factors come in pairs). For example, 256 has 9 factors because it's the square of 16, which has 5 factors (1, 2, 4, 8, 16).

Perfect Squares: To find 3-digit perfect squares, we can start by listing the perfect squares from 100 to 999: 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900.

Counting Factors: Now, we need to check which of these perfect squares have a number with 5 factors. We can do this by prime factorizing each number and counting the number of factors using the formula: (exponent of factor 1 + 1) * (exponent of factor 2 + 1) * ...

Here's a table showing the prime factorization and number of factors for each perfect square:

Perfect SquarePrime FactorizationNumber of Factors

100 2^2 * 5^29

121 11^23

144 2^4 * 3^225

225 3^2 * 5^29

256 2^89

324 2^2 * 3^415

400 2^4 * 5^225

441 3^2 * 7^29

484 2^2 * 11^29

529 23^23

676 2^2 * 13^29

841 29^23

900 2^2 * 3^2 * 5^227

Counting Numbers with 9 Factors: Finally, we count the number of perfect squares in the table that have exactly 9 factors. There are four such numbers: 100, 256, 441, and 841.

Therefore, there are four 3-digit numbers with exactly 9 factors.

Let's note that \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) is simply the perimeter of the decagon

We also know that \(radius/2 ( -1 + \sqrt {5})\) is one of the sidelenghts. Thus, we can find the perimeter easily. 

The perimeter is just

\( 10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)

Thanks! :)

A right triangle with integer leg lengths is called cool if the number of square units in its area is equal to ten times the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles?   

Plotting on Desmos, xy/2 = 10(x+y) is a hyperbola. 

I checked the curve for points where x and y are integers. 

In the positive quadrant, I found (25,100), (30,60), and (40,40).

This makes the areas 1250, 900, and 800, respectively.  Total = 2950     

I'm considering the (25,100) and (30,60) triangles the same as   

the (100,25) and (60,30) triangles, and counted them only once. 

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There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.   

Little Village ----- 4 ----- Your Town ----- 1 ----- Capital City ----- 2 ----- My Town   

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