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There are a lot of things on this site that need fixing. I don't think Admin has time for it though since he isn't here often. 

6, 8, 10 is the only one that I found.

1) By the Binomial Theorem, the constant term in the expansion of (x4+x31​)n is $(0n​)x4n+(3n​)(x31​)3=x4n+(3n​)x−9.$This term will be nonzero if and only if 4n−9 is even, which is the same as n being even. Therefore, there are 50​ three-digit positive integers n such that the expansion of (x4+x31​)n has a nonzero constant term.

Note that      [a + 1/b ] / [a -1/b ] =  [  (ab +1) / b ] /  [ (ab  -1) / b ]  =  (ab + 1) / (ab  -1)

Note that ab will be negative   and the  numerator and  denominator will always have an  absolute  value difference  of  2

The smallest possible  fraction results when   a = -2 and  b = 3  =   -5/- 7 = 5/7

The function  is  maximized when a =  -6  and  b  = 5....so....

(-6*5 + 1)  / (-6*5  - 1) =      -29   / -31   =  29 / 31

2x^2 + 5x + c =  x^2 + 7x    rearrange as

x^2 - 2x + c  = 0

This will have  at least one  real solution when  the discriminant ≥   0

So

(-2)^2  - 4(1)(c) ≥ 0

4 - 4c ≥  0

4 ≥   4c

1 ≥ c

So....it will have at least one real solution when c ≤ 1

x^2-25x+80 = -28x+75

x^2 + 3x + 5  = 0

Note that       a/b  +  b/a =   [a^2 + b^2 ] / [ab ]

Sum of roots = a + b =  -3     square both sides   

a^2 + 2ab + b^2  =  9        (1)

Product of roots = 5

So  ab = 5

2ab = 10       (2)

Sub (2) into (1)

a^2 +  10 + b^2    =  9

a^2 + b^2  =   -1

So  a/b + b/a =   -1 / 5

We have these two equations

1 = a (1 - 2)^2 + k   →    1 = a + k        (1)

-9 = a (4 - 2)^2 + k  →    -9 = 4a +k      (2)

Subtract (2) from (1)

10 = -3a

a = -10/3

k = 1 - a  = 1 - (-10/3) = 13/3

So the polynomial is

y = (-10/3)(x - 2)^2  + 13/3

And we  are trying to  find the  roots

So

(-10/3) ( x - 2)^2  + 13/3  = 0                     multiply through by 3

-10(x - 2)^2 + 13 =  0

(x - 2)^2  =  13/10            take the  positive root

x - 2 = sqrt (13/10)

x =  2 + sqrt (13/10)

Check your question.....your  root is not  correct

For each gift bag, there are three options: chocolate chip, oatmeal, or peanut butter. Since the bags are all different colors, the order in which the bags are filled matters.

Thus, there are 3×3×3=27 ways to fill the gift bags.

However, we must divide by 3! since the order in which we place the same type of cookie in a gift bag doesn't matter. (For example, if Ember puts 4 chocolate chip cookies in the first bag, it doesn't matter if they are placed in the order CCC, CCB, CBC, or BCC.)

Therefore, there are 3!27​=9 ways to distribute the 10 cookies into 4 gift bags.

The number of problems he solves on  a  normal  day  =  p*t

On the day he drinks coffee  he solves 2pt problems....so...

(4p + 7) ( t - 3)  = 2pt

4pt +7t - 12p - 21  = 2pt

2pt +  7t - 12p - 21 =  0

2pt + 7t = 12p + 21

t =  [  12p + 21 ]  / [ 2p + 7 ] 

Note that  this makes t an nteger when   p= 7  and t =  5

So.....he solves    2(7)(5) =    70 problems on the  day  he  drinks  coffee

I did not forget about y = x in my answer. In fact, here is a direct quote from my answer contradicting your claim:

"... [W]hen m = 1 and b = 0, the linear function is its own inverse. This means that \(y = x\) is its own inverse!"

Also, the question specifically concerns itself about "how many linear functions" there are with a particular property. Even if I had conveniently missed y = x, the answer would still be infinitely many, as I later concluded in my answer. 

Despite your insinuation, I am not AI. I am answering these questions genuinely, and I continue doing my best to generate high-quality answers geared towards the mathematical ability of the asker. If you have spotted a mistake in any of my previous answers, you can tell me. I will fix the mistake and give you credit accordingly for spotting the error.

Call the area of the base =  B

Call the area of  each triangular face B / 2

So 

 B + 4(B / 2)  = 432

B + 2B =  432

3B = 432

B = 144

So ...the side of  the base = sqrt (144)  =  12

And the area  of  each triangular face  =   B / 2 =  72

The slant height of the  pyramid = 

area of triangular face  = (1/2) (slant height) (base side)

72 = (1/2)(slant height)(12)

144 = (slant height) (12)

12 = slant height

Height of pyramid =  sqrt [ slant height^2 - (base length / 2)^2 ] = sqrt [ 12^2 - 6^2] =

sqrt (108)  = sqrt (36 * 3)  = 6sqrt3

Volume of  pyramid = (1/3) (base area) (height)  = (1/3)(144)(6sqrt 3)  =  288sqrt 3  units^3

One equation , three unknowns....we may have multiple solutions !!!

WolframAlpha actually shows three different solutions :

https://www.wolframalpha.com/input?i=6xyz+%2B+30xy+%2B+3xz+%2B+8yz+%2B+15x+%2B+44y+%2B+8z+%3D1342

We can find the inverse of an arbitrary linear function and find its inverse and determine how many instances there are when the linear function is an inverse of itself.

\(y = mx + b\) is the typical form of a linear function. Let's find its inverse.

1) Swap the variables x and y.

\(x = my + b\)

2) Solve for y-variable.

\(x = my + b \\ my = x - b \\ y = \frac{1}{m}x - \frac{b}{m}\)

Now we know that for an arbitrary linear equation \(y = mx + b\), \(y = \frac{1}{m}x - \frac{b}{m}\) is the inverse. When are these equations equal? These equations are equal when the coefficient of x and the constant term are the same! This leads to a system of equations.

\(m = \frac{1}{m}; b = -\frac{b}{m} \\ m = \frac{1}{m} \\ m^2 = 1 \\ m = 1 \text{ or } m = -1\)

There are two possibilities for m. Now, let's find the corresponding values for b that make linear functions their own inverse.

\(\text{Let } m = 1: \\ b = -\frac{b}{m} \\ b = -b \\ 2b = 0 \\ b = 0\)

This mean that when m = 1 and b = 0, the linear function is its own inverse. This means that \(y = x\) is its own inverse!

Now, let's try find the other value of b when m = -1.

\(\text{Let } m = -1 \\ b = -\frac{b}{m} \\ b = -\frac{b}{-1} \\ b = b\)

The left-hand side and the right-hand side of the equation are the same equation. This means that the value of b can be anything, and the equation will hold true. In other words, \(y = -x + b\) where b is any real number has the property that the linear function is its own inverse. This means that y = -x and y = -x + 1, y = -x + 3, y = -x + 1000000000000 are all inverses of themselves.

This means there are infinitely many linear functions that have inverses of themselves.

C (48 , 8)  =  377,348,994  different  choices

Convert inches to  feet :

108 / 12  =  9 ft

9 ft /  2 =   4.5  ft  = the length of each piece

I am confused with this answer. Why is it that the interval that satisfies neither inequality is the union of the intervals found? I would expect the interval that satisfies neither of the inequalities to be the intervals of real numbers not contained within the original interval. That would be \((-1, 6)\). This is the interval of real numbers that is greater than -1 and less than 6.

ElectricPavlov's method is perfectly valid, but there is a method that avoids having to do arithmetic with square roots, which can become quite messy. I mostly thank Cphill for presenting Vieta's formula in a previous answer. Make sure the quadratic equation is in standard form first.

\(3x^2 + 4x - 9 = 2x^2 - 6x + 1 \\ x^2 + 10x - 10 = 0\)

By Vieta's formula, we can deduce the product of the roots is the constant term, and the sum of the roots is the opposite of the coefficient of the x-term. This means that \(r_1 + r_2 = -10 \text{ and } r_1r_2 = -10\). Our ultimate goal is to find \(r_1^2 + r_2^2\).

\(r_1 + r_2 = -10 \\ r_1^2 + 2r_1r_2 + r_2^2 = 100 \\ r_1^2 + 2 * -10 + r_2^2 = 100 \\ r_1^2 + r_2^2 = 120\)

We have found the sum of the square of the roots!

First, simplify the quadratic into its standard form. This will allow you to identify information about the roots of the equations.

\(2x^2 + 6x - 14 = x^2 - 8x + 2 \\ x^2 + 14x - 16 = 0\)

We can also expand the product of the two binomials. This will be important shortly, as you will see.

\(\begin{align*} (2a - 3)(4b - 6) &= 8ab - 12a - 12b + 18 \\ &= 8ab - 12(a + b) + 18 \end{align*}\)

We can use Vieta's formula to find the product of the roots and the rum of the roots. For a quadratic, the product of the roots is the constant term, and the sum of the roots is the opposite of the coefficient of the x-term.

\(ab = -16, a + b = -14 \\ \begin{align*} 8ab - 12(a + b) + 18 &= 8 * -16 - 12 * -14 + 18 \\ &= -128 + 168 + 18 \\ &= 58 \end{align*} \)

0.13 is written to the hundredth place, which means that its corresponding fraction can be represented as a fraction with 100 as the denominator.

\(0.13 = \frac{13}{100}\)

Here are a few more examples that may make this concept make sense.

0.7 is written to the tenth place, which means that its corresponding fraction can be represented with 10 as the denominator.

\(0.7 = \frac{7}{10}\)

0.111 is written to the thousandth place, which means that its corresponding fraction can be represented with 1000 as the denominator.

\(0.111 = \frac{111}{1000}\)

I tried to read SpectraSynth's answer, and I was unable to make sense of it. I arrived at a different answer than SpectraSynth, so I will present my method, and you can figure out which answer is the correct answer.

Our ultimate goal is to find the area of Rectangle ABCD. If we knew the length and the width of the rectange, then finding the area would be trivial because \(A_{\text{rect}} = l \times w\) where \(l\) is the length and \(w\) is the width. As such, I will label \(AB = l\) and \(AD = w\). 

By definition, we know that Rectangle ABCD is a quadrilateral with four right angles. This observation lead me to realize that \(\triangle \text{ABE} \text{ and } \triangle \text{ADF} \text{ and } \triangle \text{ECF}\) are all right triangles. Because they are right triangles, finding a relationship between the lengths of the triangle legs and the area will be easier to deduce. \(\overline{BE}\) is the height of \(\triangle \text{ABE}\) and \(\overline{\text{DF}}\) is the height of \(\triangle \text{ADF}\). I will let \(BE = x \text{ and } AD = y\). With this information, I can already generate two equations that relate the triangle legs and the area. I simplified these equations as much as I could.

We can employ a similar strategy for \(\triangle \text{ECF}\). However, it is almost always best to relate new variables to information that you already have. Because figure ABCD is a rectangle, we can relate the \(\text{EC} \text{ and } \text{EF}\) to other lengths we already know.

Now that we have related it to variables we have already defined, we can use the area formula for rigth triangles to find more information.

\(A_{\triangle{\text{ECF}}} = \frac{1}{2} * \text{EC} * \text{CF} \\ 10 = \frac{1}{2}(w - x)(l - y) \\ 20 = lw - wy - lx + xy\)

We can already simplify this complicated equation by recognizing that we know from previous equations that \(lx = 10 \text{ and } wy = 10\).

\(20 = lw - 10 - 10 + xy \\ lw + xy = 40\)

This equation is close to solved because we only need to solve for \(lw\), which is equivalent to the area of Rectangle ABCD. We can use previous equations to substitute in an expression for \(xy\). We can multiply previous equations together to find a relationship in which we can ultimately use substitution.

\(lx = 10; wy = 10 \\ lxwy = 100 \\ xy = \frac{100}{lw}\)

Now, we can substitute this relationship into the equation and solve for \(lw\).

\(lw + xy = 40; lw = \frac{100}{lw} \\ lw + \frac{100}{lw} = 40 \\ (lw)^2 + 100 = 40lw \\ (lw)^2 - 40(lw) + 100 = 0\)

The equation simplifies to a quadratic equation, which I will solve with the trusty quadratic formula.

\({\color{red}1}(lw)^2 {\color{blue} - 40}(lw) {\color{green}+ 100} = 0 \\ lw_{1, 2}=\frac{-({\color{blue}-40}) \pm \sqrt{({\color{blue}-40})^2 - 4 * {\color{red}1} * {\color{green}100}}}{2 * {\color{red}1}} \\ lw_{1, 2} = \frac{40 \pm \sqrt{1600 - 400}}{2} \\ lw_{1, 2} = 20 \pm \frac{20\sqrt{3}}{2} \\ lw = 20 - 10\sqrt{3} \text{ or } lw = 20 + 10\sqrt{3} \\ lw \approx 2.6795 \text{ or } lw \approx 37.3205\)

The quadratic yields two different answers, and they both seem logical. Are there two answers? Upon studying the diagram carefully, I rejected \(lw = 20 - 10\sqrt{3}\) because Rectangle ABCD has a minimum area of \(5 + 5 + 10 = 20\) because the rectangle contains three triangles of which we know the area and one non-right triangle of which we do not know the area. Since \(20 - 10\sqrt{3} < 20\), I rejected this answer.

Therefore, \(A_{\text{Rectangle ABCD}} = lw = 20 + 10\sqrt{3} \approx 37.3205\)