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The points where the graphs intersect are points that lie on both graphs. Therefore, we can set \(x^3+x^2-3x+5=x^3+2x^2 \). Combining like terms, we get \(x^2+3x-5\). By the quadratic formula, the roots of this equation are \(\frac{-3\pm\sqrt29}{2}\). We have to plug these values back into one of the two equations to figure out the y coordinate. I'll let you figure that out (sry I don't have scratch paper rn).

Feel free to tell me if I did anything wrong! :D

First, we look at 1000 mod 21 and 2000 mod 21. 1000 is congruent to 13 mod 21, and 2000 is congruent to 5 mod 21. We are trying to find how many numbers between 1000 and 2000 are congruent to 3 mod 21. The greatest number is 21*95+3=1998 and the least number is 21*48+3=1011. Because it cycles every 21 numbers, there are 95-48+1=\(48\) numbers that satisfy this property

Feel free to tell me if I did anything wrong! :D

If we rationalize the denominator, we get \(-\frac{\sqrt2-\sqrt3}{1}-\frac{\sqrt2+\sqrt3}{1}\). If we combine like terms, we get \(-2\sqrt2\).

Feel free to tell me if I did anything wrong! :D

I just wanted to note that booboo44's solution may be incorrect. First off, there are 6 choose 2 = 15 possible outcomes, not 30. This is because cases will be counted twice. In booboo44's solution, AB and BA are treated as different cases, when they are two equal cases. The favorable cases should also be 12, not 16. There may be more mistakes later on in the solution (but I won't be covering them). If all calculations are done right, the answer should be 8/15. (I created a solution but apparently it's still being moderated)

This is for educational purposes only. I do not want to start a fight with anyone.

Feel free to tell me if I did anything wrong! :D

If we combine like terms, we get ab-9a-5b+__. Then, we can factor the expression as a(b-9)-5b+__. Because -5b+__ need to be factorable by b-9, the blank must be 45. If we plug it back in, we can factor it into (a-5)(b-9). Therefore, the blank is 45.

Feel free to tell me if I did anything wrong! :D

Analyzing f(x)⋅g(x)

We are asked to find several properties of the product of two polynomials, $f(x) = -x^2+ 8x - 5$ and $g(x) = x^3 - 11x^2 + 2x$.

Finding the Product:

The product f(x)⋅g(x) can be found using the distributive property or polynomial multiplication techniques. Here, we'll use the distributive property:

f(x) * g(x) = (-x^2 + 8x - 5) * (x^3 - 11x^2 + 2x)

Multiplying each term of f(x) by each term of g(x) and combining like terms will result in a fourth-degree polynomial.

Properties of the Product:

Leading Term:

The leading term in a polynomial is the term with the highest degree. In the product, the highest degree term will come from multiplying the highest degree terms of f(x) and g(x). These terms are -x^2 and x^3, respectively. Their product is -x^5. Therefore, the leading term of f(x) * g(x) is -x^5.

Leading Coefficient:

The leading coefficient is the coefficient of the leading term. In this case, the leading term is -x^5, and its coefficient is -1. Therefore, the leading coefficient is -1.

Degree:

The degree of a polynomial is the highest exponent of the variable. Since the leading term has a degree of 5, the degree of f(x) * g(x) is 5.

Constant Term:

The constant term is the term that doesn't include any variable (x). To find it, we identify the terms in the product that don't have any x. Multiplying the constant term of f(x) (-5) by the constant term of g(x) (2) will result in the constant term of the product. Additionally, there might be other terms that don't have x due to the product of other terms. After multiplying and combining like terms, you'll find the constant term.

Coefficient of x^2:

The coefficient of x^2 is the number multiplying the term with x^2. This can be found by identifying terms in the product that have x^2. Multiply terms in f(x) that contain x by terms in g(x) that contain x^2, and vice versa. Combine like terms to find the coefficient of x^2.

Finding these properties requires multiplying f(x) and g(x) and performing some term manipulation. However, the steps outlined above should guide you in finding each property.

At first glance, it might seem like there are 5 options for each side and so a total of 5 * 5 * 5 = 125 ways to color the triangle. However, this overcounts the number of colorings because rotating or reflecting the triangle can create what seems like a different coloring.

We can consider two cases:

Case 1: All three sides are the same color.

There are 5 choices Sam can make for this single color (since all three sides are the same).

Case 2: Two sides are the same color and the third side is a different color.

Here, we need to consider how many ways Sam can choose the two colors and how many ways he can arrange them.

Choosing the colors: There are 5 choices for the color used twice and 4 remaining choices for the different color (since one color is already chosen for the two sides). So, there are 5 * 4 = 20 ways to choose the colors.

Arranging the colors: It might seem like there are 2 ways to arrange the colors (one color for two sides and another for the remaining side). However, rotating the triangle flips the arrangement. So, if we consider both rotations as the same coloring, then there is only 1 way to arrange the colors.

Therefore, for Case 2, the total number of colorings is 20 (choosing colors) * 1 (arranging colors) = 20.

Total Colorings:

Adding the number of colorings for both cases:

Total Colorings = Case 1 + Case 2 = 5 (all same color) + 20 (two same, one different) = 25

So, Sam can color the triangle in 25 different ways.

Combining all like terms and expanding, we get that

\(6c+3=219c-27\)

Isolating c to one side, we get

\(-213c=-30\)

Dividing both sides by -213, we get

\(c=\frac{10}{71}\)

So 10/71 is our only answer. 

Thanks! :)

The sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers that are neither perfect squares nor perfect cubes.

One way to find the 1000th term is to simply count the number of perfect squares and perfect cubes less than a very large number, then subtract that number from the very large number itself, assuming it is not a perfect square or cube itself.

The 33rd perfect square is 332=1089

The 12th perfect cube is 123=1728

Since 1089<1728, there are more perfect squares than perfect cubes less than 1728.

We know 1 is a perfect square and perfect cube, so we subtract 2 from the count.

Therefore, there are 33−2=31 perfect squares and cubes less than 1728.

Since 1728 is not a perfect square or cube, the 1728th term in Cai's list is 1728.

Following this logic, to find the 1000th term, we can look for the smallest perfect square greater than 1000.

The 32nd perfect square is 322=1024

There are 32−2=30 perfect squares and cubes less than 1024.

Since 1024 is a perfect square, the 1000th term in Cai's list is not 1024. However, we can see that the difference between the 1000th term and the 1024th term is less than or equal to the difference between the 999th term and the 1000th term (which is 1).

Therefore, the 1000th term must be 1 less than the 32nd perfect square, which is 322−1=1023​.

We basically have all the information needed to solve this problem. 

First off, let's note that 

\((x+y)^3 = x^{3}+3x^{2}y+3xy^{2}+y^{3}\)

Simplifying this, we get

\(8=5+6xy\\ xy=1/2\)

Now, let's note that \((x+y)^2 = x^2+2xy+y^2\)

This contains x^2+y^2, so let's isolate that. We get

\(x^2+y^2 = (x+y)^2 - 2xy\)

We already have all the values shown for us to find x^2+y^2. Plugging in 2 and 1/2, we get

\(x^2 + y^2 = 4 - 1 = 3\)

So 3 is our final answer. 

Thanks! :)

There actually is no minmum for y. 

The range for y is \(f\left(x\right)\le \frac{1}{4}\) as there is no limits to how small it can be. 

There is a maximum of 1/4, as the max point is

\(\left(1,\:\frac{1}{4}\right)\)

Thanks! :)

We can split this equation into 2 equations. Since if either them equal 0 the equation will be true, we just set both quadratics to equal 0. 

We have

\(2x^2+4x+2=0\quad \mathrm{or}\quad \:3y^2-6y+3=0\)

Solving the first equation, we have

\(2(x^2+2x+1)=0\\ 2(x+1)^2=0\\ x=-1\)

Solving the second equation, we have

\(3(y^2+-2y+1)=0\\ 3(y-1)^2=0\\ y=1\)

So there are an infinite amount of ordered pairs for x and y as long as either \(x=-1\) or \(y=1\)

So infinite is our answer...

Thanks! :)

In that case, we can do the same with 7 and 8. 

We have 

\(\binom{1 + 4 - 1}{4 - 1} = \binom{4}{3} = 4\)

and

\(\binom{0 + 4- 1}{4 - 1} = \binom{3}{3} = 1\)

So there are 10+4+1 = 15. 

So our answer is 15/495. 

Thanks! :)

That's incorrect. Thanks, tho!

The new area of Square A is (2009 +x)^2, while the new area of Square B is (2009 - x)^2. The difference is \(\begin{align*} &(2009+x)^2-(2009-x)^2\\ &\qquad=(2009+x+2009-x)(2009+x-2009+x) \\ &\qquad=(2\cdot 2009)(2x) \end{align*}\)

For this to be at least as great as the area of a 2009 by 2009 square, \(2(2009)2(x)\geq 2009^2\Rightarrow x\geq \boxed{\frac{2009}{4}}.\)

We want bc to be the biggest, ac to be the second biggest, and ab to be the smallest. 

Obviously, c has to be the biggest number and b has to be the second biggest number. 

Thus, c is 3, b is 2, and a is 1. 

We have

\((2*1) + 2(1)(3) + 3(3)(2)\\ 2+6+18\\ 26\)

So our answer is 26. 

Thanks! :)

Let's convert the each number in the fraction to base 25. 

One always stays the same, so converting 288 to base 25, we have

\((BD)_{25} = (11 × 25^1) + (13 × 25^0) = (288)_{10}\)

This means that

\(\frac{1}{288}_{10} = \frac{1}{BD}_{25}\)

Now, we do base division. We get that

\(\frac{1}{BD}_{25} = 0.\overline{02468ACEGIKN}\)

As you can see, 1/288 in base 25 is repeating. 

So repeating is our answer. 

Thanks! :)

Let's split this into two inequalities and solve both to find what x is. 

We have

\(64 \leq n^2\)

and 

\(n^2 \leq 100+20n\)

Let's solve the first equation. Square rooting both sides, we get

\(n \geq 8\) or \(n \leq -8\)

Solving the second equation, we get that

\(n^2-20n-100 \leq 0\\ 10\left(1-\sqrt{2}\right) \leq n\leq10\left(1+\sqrt{2}\right)\)

Merging overlapping intervals, we have

\(8\le \:n\le \:10\left(1+\sqrt{2}\right)\)

This rounds to 

\(8\le \:n\le \:24.14\)

So our answer is 17. 

Thanks! :)

First, let's combine all like terms in the equation. We have

\(4x^{4}{-10x^{2}}\)

Factoring this equation, we get that

\(2x^{2}\left(2x^{2}-5\right)\)

Now, we can graph this. We have that

So the minimum value is -6.25. 

So our final answer is -6.25 

Thanks! :)