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Subtract \(3 \) from both sides:

\(-3x^2+24x \leq 0\)

Factor the equation by \(3x\):

\(3x(-x+8) \leq 0\)

Both \(3x\) and \(-x+8\) have to be less than or equal to \(0\). This means your answer is \(\color{red}x \leq 0\) or \(\color {red}x \geq 8\)

\({{5 \over 8} * {1 \over 5}} = {1 \over 8}\)

So, \(1 \over 8\) of the tractors are in working conditions.

\({{{5 \over 8 } + {3 \over 8} } \over 2} = {1 \over 2}\) 

This means that Mary read \(1 \over 2\)of the two books.

the 51st letter is a

the 102nd letter is z

the 2091st letter is a

the 2101st letter is j

i think...

Hi! Nice to meet you. :)

Hello guest!

wins = \(2 + 2n\)

losses = \(13 + n\)

This means the equation is \({{2 + 2n} \over 13 + n} = 0.4\)

Multiply each side by \(12 + n \):

\(2 + 2n = 5.2 + 0.4n\)

Subtract \(2\) from both sides:

\(2n=3.2+0.4n\)

Subtract \(0.4n\) from both sides:

\(1.6n=3.2\)

Divide each side by \(1.6 \):

\(n=2\)

This means that after \(2\) weeks, they would have won \(40\)% of their games. 

x + y + z = 1. Since xyz is in the denominator, we want xyz to be the largest value possible.

The largest value possible of xyz is only if x = y = z, so x = y = z = 1/3.

\(1\over{({1\over3})^2}\) = \((x + y + z)\over{xyz}\)

Thus, the minimum value of \((x + y + z)\over{xyz}\) is 27.

(x - sqrt(125))²   =  x^2 -10 sqrt5 + 125

(x+ sqrt (125))² =   x^2 +10 sqrt5 + 125

This is a Venn Diagram question.

If 37 students only take spanish, then there are 60 students who either take only Chinese or both. If there are 5 more students in Chinese than in Spanish, then there are 37 + 5 more students who only do Chinese.

This is true because no matter how many students take both Chinese and Spanish, it doesn't contribute to the fact that there are 5 more students who take Chinese than Spanish. Altogether there are 79 students who take one language, so there are 97 - 79 = 18 students who take both languages.

Hi Asinus! You may have misread, but angleAEF = 10angleBAC - 22 degrees, not AEF = 10angleBAC. :)

By alternate interior angles, angleAEF is congruent to angleDAE. 

Since CAE is a straight line, then it has a straight angle. Therefore angleBAC + angleBAD + angleDAE = 180degrees.

angleDAE = angleAEF = 10angleBAC - 22degrees.

Since angleBAC = angleBAD because BA is an angle bisector, then we can simplify the equation to:

angleBAC + angleBAC + 10angleBAC - 22 = 180degrees.

To simplify:

12 * angleBAC = 202degrees.

To simplify:

angleBAC = (101/6)degrees.

Since we are looking for angleDAC, and we also know 2 * angleBAC = angleDAC, then angleDAC = \(101\over3\)degrees.

 What is angle DAC in degrees?

Hello Guest!

\(AEF+BAC+BAD=180^o\\ 12BAC=180^o\\2BAC=\color{red}DAC=30^o\ error \\ \)

\(10BAC-22^o+DAC=180^o\\ 5DAC-22^o+DAC=180^o\\ 6DAC=202^o\\ \color{blue}DAC=33\frac{2}{3}^o\)

  !

P - 1/4P ==3/4P non-dancers.

3/5*3/4P ==9/20P women non-dancers.

3/4P - 9/20P ==3/10P men non-dancers.

1/4P - 36 == women dancers.

3/10P ==1/4P - 36 + 86, solve for P

P==1000 people at the party. 

If the square root of \((63 - 3\sqrt{x})\) is an integer, then it has to be a perfect square.

Here are perfect squares under 63:

49 = 7^2 

36 = 6^2

25 = 5^2

16 = 4^2

9 = 3^2

4 = 2^2

1 = 1^2

0 = 0^2

Since there are 8 perfect squares under 63, then there are 8 real values of \(x\) where the \(\sqrt{(63 - 3\sqrt{x})}\) is an integer.

If the angles form an arithmetic sequence, then we can create a variable for the difference in angles.

If the largest angle is 128 degrees, then the second largest angle is 128 - x degrees, the third largest is 128 - 2x degrees, and the smallest angle is 128 - 3x degrees. 

We also know the interior angle sum of a quadrilateral adds up to 360 degrees.

Therefore, 128 + 128 - x + 128 - 2x + 128 - 3x = 360. 

Then we get x = 76/3 

Since we are looking for the second largest angle, it is just 128 - 76/3. 

Thus, the second largest angle is 308/3 degrees or 102.666666 degrees.

nice answer, just note that the question asks for the entire solution set, which is 360n±51.13, where n is an integer.

16 + 8 - 5 = 19

-12 thru  -1   cancel  out the 1 thru 12        then there is only  13 left   

total = 13

832 , 842 , 852 , 932 , 942 , 952 , Total =  6 such integers.

 N=2^4*3^8*5^2*7^2==128,595,600

128595600 = 2^4 * 3^8 * 5^2 * 7^2.  As you can see, it has 4 DISTINCT prime factors{2, 3, 5, 7}

Counting the repeats, then just add the exponents up:4 + 8 + 2 + 2 ==16 prime factors.

If you want to know the number of "DIVISORS", then it has:

[4+1] * [8+1] *  [2+1] * [2+1] ==5 x 9 x 3 x 3 ==405 DIVISORS.

The first part of the expression can be written like this:

a(2a  +  b) ==2a^2  +  ab. Now, look at the 2nd part of the expression: 2a^  +  ab. Do you see any difference between the two? They are exactly the the same! So, no matter what 2 different numbers you enter, primes or not, you will always get "0". Unless you made a mistake in typing the expression.

4(x + 7)(2 - x) + 2x^2 + 8x - 3     simplify to

-4x^2 -20x +56    + 2x^2 + 8x-3

-2x^2 -12x + 53                  max will occur at x =   -b/2a  =  - -12/((2)(-2) = - 3     

                                                  use this 'x' in the equation to calculate the max value  (the 'y' value)

Slope, m = 3/-3 = -1

 y = -x + b

11 = -1+ b      b = 12

y = - x + 12

x+y = 12

x/12 + y/12 = 1     < ========   a = 12