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\(y=2(3^x)\qquad and \qquad y=6(2^x)\\ so\\ 2(3^x)=6(2^x)\\ \frac{3^x}{2^x}=3\\ 1.5^x=3\\ log1.5^x=log3\\ xlog1.5=log3\\ x=\frac{log3}{log1.5}\\ x\approx 2.7095\)

You can take if from here.

BuilderBoy: Do you think you can solve this question using combinatorics? Give it a try. Thanks.

How did you get to that answer? Its close to mine but I am not sure if mine is right.

This is what I got.

2(3^x) = 6(2^x)

(3/2)^x = 6/2

(3/2)^x = 3

log(3/2) * x = log3

x = log(3)/log(3/2)

x = 2.7

Plug it into either one:

2(3^2.7095) = 6(2^2.7095)

= 39.2

Point of intersection (x,y) = (2.7, 39.2)

Well, if there is only 3 games, and 1/5 probability for each, I believe you just do:

1/5 * 1/5 * 1/5

= 1/125 probability that the Bears actually win the tournament.

However, if I am wrong, please tell me, as I would be happy to learn what my mistake was and how I could have corrected it!

I don't see "the 5 boxes below"...

In future, use this calculator: Wolfram|Alpha: Computational Intelligence

Remember this: 2,5 in Russian is written as: 2.5 in English.

Yes, the answer of 10.89% is correct.

If x = 12, then your equation simplifies to:

$$36 + y = a$$

and

$$24 + 5y = 2a$$

We can substitute a for 36 + y:

$$24 + 5y = 72 + 2y$$

$$3y = 48$$

$$y = 16$$

We now know that y is 16, and we substitute that into our first equation:

$$36 + 16 = a$$

$$52 = a$$

$$a = 52$$

So the value of a is 52.

If I am wrong, please tell me as I would be happy to fix it!

Oh, right, you could also solve it using discriminants. Didn't even think of that 

Pretty sure your answer is just a repeat. Check out this one:

Here's another way: 

The equation simplifies to \(x^2 + 9x + c = 0\)

The equation has at least 1 real root when the discriminant \(b^2 - 4ac \geq 0\)

Substituting in what we have gives us \(81 - 4c \geq 0 \), meaning \(c \leq 20.25\), so the maximum value of c is \(20.25 = \color{brown}\boxed{81\over 4}\)

I don't see a grid either.

What image?

Alright, so we're going to write a system of equations and define the variables:

d = the number of drinks Jordan buys

t = the number of tacos Jordan buys

We know that he bought 4 more tacos than drinks so:

$$t = 4 + d$$

Our next equation is:

$$2.50 \cdot d + 2 \cdot t = 26$$

Since t = 4 + d, we substitute this into our equation:

$$2.50 \cdot d + 2 \cdot (d + 4) = 26$$

We simplify:

$$2.50d + 2d + 8 = 26$$

Subtract 8 from both sides:

$$2.50d + 2d = 18$$

$$4.50d = 18$$
 

We divide by 4.50

$$d = 4$$

So we know that he bought 4 drinks, and when we do d + 4 = 8 tacos.

If anything is wrong, please tell me as I would be happy to help!

Hmmm...

We have the equation:

$$2x^2 + 5x + c = x^2 - 4x$$

We subtract x^2 from both sides:

$$x^2 + 5x + c = -4x$$
We add 4x to both sides:

$$x^2 + 9x + c = 0$$

Now, I want to clarify one thing, when finding the largest product, there is a pretty easy way to do this with intuition.

Try multiplying:

$$7*\cdot3 = 21$$

$$6\cdot4 = 24$$

$$5\cdot5 = 25$$

When doing this similar multiplication process with other numbers, you will realize that the closer the numbers are, the greater their product is:

So, when we want the max value of our constant, we have to use the closest numbers together.

We also notice that our two numbers (r and s) have to equal 9. Since we need the closest numbers, then r = s.

$$2r = 9$$

$$r = 4.5$$

So our equation would be in the form:

$$(x + 4.5)(x + 4.5) = x^2 + 9x + 20.25$$

Therefore the greatest constant value is 20.25.

Expressing this answer in a fraction can either be 20 1/4 or 81/4

Please tell me if I am incorrect as I would be happy to fix it!

I'm pretty sure this is an illogical question, because I don't see any numbers, and you have unnecessary spaces where numbers should be. Please fix that...

I'm new here so maybe I'm missing something, but I don't see any grid below, is there something wrong with your question, or am I missing something?