NotThatSmart

First, let's set up variables to solve this question. 

Let's let \(d = a_2 - a_1\)

The conditions in the problem give us the system of equations

\(\begin{cases} a_1 + d = -1\\ a_1 + 3d = \dfrac13 \end{cases}\)

Now, solving this equation, we get that

\(d = \dfrac23\) and \(a_1 = -\dfrac53\)

So the final answer is -5/3. 

Thanks! :)

OK, sure! no problem. I'm not 100% sure of my answer, but I'll give it my best. 

Let's use case work to solve this question. I'm assuming that children cannot choose to not have ice cream. 

First, let's say 3 children chose flavor A. 

There are \(\binom 63 ​\cdot 2^ 3 =20⋅8=160\)

This is because we need to choose 3 out of the 6 children to have one flavor, and the other 3 can choose one of the other 2. 

The same applies to flavors B and C. 

Thus, we have \(160 \cdot 3 = 480\)

I THINK 480 is the answer. Disclaimer, my counting skills are sub-level...but yeah...

Thanks! :)

You are missing a value. 

a single scoop of any of  flavors of ice cream

How many flavors of ice cream. 

PLease input the right amount, and I'll help you then :)

I know this is a late response, but I just realized the answer to this question. 

Now, even though the blue checkmark has been given, signifying the right answer, I don't believe the anwer was correct. 

Let me explain my reasoning. 

First off, the final answer achieved in the previous response (no knock, you almost had the right answer) was

\(\left(a\:+\:b\:+\:c\right)\left(ab\:+\:bc\:+\:ca\right)\) This simplifies to \(a^2b+a^2c+ab^2+b^2c+bc^2+ac^2+3abc\)

This can't be right because note that in the original problem, there are powers of 3 for variables a, b, c, while in this, there are none. 

Now, let's me explain my tactic. Hang with me, it's quite tedious. 

Now first off, let's set a multivariable function to deal with this question, 

First, let's let \(f(a,b,c) = (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3\). (This will come into handy later)

This following step is not necessary, as you can just easily follow through, but expanding everything, which is quite tedious, we get

\(3a^3b^2c+3a^2b^3c+3a^3bc^2+3a^2bc^3+3ab^3c^2+3ab^2c^3+6a^2b^2c^2\)

Now, let's note that every term is divisble by 3abc. Factoring this out, we get

\(f(a, b, c,)=3abc(b^2c+a^2c+a^2b+ab^2+2abc+bc^2+ac^2)\)

The following steps are complicated, but I do believe this was explained in an AOPS course, which is where this problem came from.

(I know because I had the same problem while taking the course myself, it was a writing problem)

Let's note the following ideas. 

If we let\( c=-a\), we find that \(f(a,b,-a) =0\), meaning that a+c is a factor of \(f(a,b,c)\)

If we let \(-a = b\), we also find that \(f(a, -a, c) = 0\). This also means that a+b is a factor. 

If we let b = -c,  \(f(a, -c, c) = 0\). This also means that b+c is a factor. 

This also means that \((a+b)(b+c)(c+a)\)should be in the final factorization. 

Wait a sec! Notice something really quickly!

\((a+b)(b+c)(c+a) = b^2c+a^2c+a^2b+ab^2+2abc+bc^2+ac^2\), which means that we can replace what we had in parethensis in the original facorization with \((a+b)(b+c)(c+a)\)

Thus, our final factorization is \(3abc(a+b)(b+c)(c+a)\)

Thus, our final answer is \(3abc(a+b)(b+c)(c+a)\)

Sorry for the late response. This should be correct, but correct me if it is not. 

Thanks! :)

First, let's expand out everything. I'm not going to show the steps, but we eventually get

\(2z^4-32z^3+204z^2-608z+706=-8\\2z^4-32z^3+204z^2-608z+714=0\)

Now, since every number is divisible by 2, we get factor out 2 to get

\({2\left(z^{4}-16z^{3}+102z^{2}-304z+357\right)}=0\)

Now, note that \(z^{4}-16z^{3}+102z^{2}-304z+357=(z^{2}-8z+17)(z^{2}-8z+21)\)

This means that we have the equation

\(2\left(z^{2}-8z+17\right)\left(z^{2}-8z+21\right)=0\)

Since the equation is equal to 0, we get two equations. We have

\(z^{2}-8z+17=0,\qquad z^{2}-8z+21=0\)

Using the quadratic equation on both formulas, we find 4 values of z. 

We have

\(z=4+i\\ z=4-i\\ z=4+\sqrt{5}\cdot (i)\\ z=4+\sqrt{5}\cdot (-i)\)

This was a very basic explenation of the problem. 

I can go in dephth more if you want to. 

Thanks! :)

Thanks for the confirmation, CPhill. 

:) :) ;)

First, let's note that if there are no real solutions, then that means the descriminant is less than 0, 

Simplifying the quadratic equation a bit, we find that

\(-15x^2-kx+53=0\)

Since the descriminant is b^2-4ac, we have the equation

\(k^2+3180<0\)

This yields \(k^2<-3180\)

Note that this isn't possible. k^2 must be greater or equal to 0, so it cannot be equal to -3180. 

I might have done something wrong...not sure, but I think it's impossible

Thanks! :)

(b) First, we let's set an equation. 

Since they both add up to 50, we have the equation

\(\:\frac{30}{100}x+\frac{60}{100}y=54\) where x is the amount of blue potion and y is the amount of red potion. 

Since they add up to 100mL, we also have the equation \(x+y=100\)

From the second equation, we isolate x to find that \(x=100-y\)

Plugging this value of x into the first equation, we get

\(\:\frac{30}{100}(100-y)+\frac{60}{100}y=54\)

Now, I won't show the steps, but solving this equation, we get that

\(y=80\)

This means that \(x=20\)

Thus, we find that we need 80mL of red potion and 20mL of blue potion. 

Thanks! :)

(a) Let's make some obersvations about this problem first. 

First off, if there's 500 mL, then we have \(500 * 30\% = 150 mL\) of magic maple syrup. 

Setting this up as a fraction, we have \(\text{Percentage of Syrup} = \frac{150}{500}\)

Now, since 60% of red potion is syrup, then we have the equation

\(\frac{40}{100} = \frac{150+\frac{6}{10}x}{500+x}\) where x is the amount of mL of red potion we must add. 

Thus, now we solve for x. 

Cross multiplying the equations, we get

\(100000+200x=75000+300x\\-100x=-25000\\x=250\)

Thus, she must add 250mL of red potion. 

Thanks! :)

I believe the rhombus is a square for main reasons. 

For one thing, the opposite angles of a rhombus are 90 degrees, so we have two 90 degree angles.  

The adjacent angle also should be 90 degrees, so all 4 angles are 90 degrees. 

Therefore, we just have a square. 

We have \(12*12=144\)

So the area is 144. 

Thanks! :)