NotThatSmart

First, let's use the Law of Cosines. It states that \( cos PMQ = - cos PMR \)

Thus, we can write the equations

\(PQ^2 = QM^2 + PM^2 - 2(QM * PM) * (-cos PMR) \\ PR^2 = RM^2 + PM^2 - 2 (RM * PM) * ( cos PMR)\)

Plugging in the values we already know from the problem, we get

\(5^2 = 5.5^2 + PM^2 + 2(5.5 * PM)cos(PMR) \\ 8^2 = 5.5^2 + PM^2 - 2(5,5 * PM) cos (PMR)       \)

Now, add these two equations. We get

\(5^2 + 8^2  = 2 * 5.5^2 + 2PM^2 \\ 89 = 60.5 + 2PM^2 \\ 28.5 / 2 = PM^2 \\ 14.25 = PM^2 \)

Thus, we have \(\sqrt{14.25}  = PM ≈  3.77\)

The answer is 3.77. 

Thanks! :)

First, let's simplify and combine all like terms. 

After doing that, we get \(3x^2-kx+33=0\)

Now, in order for the equation to have no real solutions, the descriminant must be less than 0. 

The descriminant is \(b^2-4ac\), so we have the equation \(k^2 - 4(33)(3) < 0\)

Since we are trying to find the largest, we set the two to equal each other and get

\(k^2 - 396 = 0\\ k^2 = 396\\ k \approx 19.89974\)

Rounding down, the largest k can be as an integer is 19. 

So our answer is 19. 

Thanks! :)

First, let's combine all like terms and put this in the form of a quadratic equation. We have

\(3x^{2}+4x+10\)

Now, the descriminant is in the form of \(b^2-4ac\) for qudratics in the form \(ax^2+bx+c\)

Thus, we have

 \(4^2-4(3)(10)\\ = 16 - 120\\ -104\)

So the answer is -104. 

Thanks! :)

Good answer by EP here

I do not think this is a geometric series. 

We have 

\(0.0000324/0.036=0.0009\)

\(0.036/0.4=0.09\)

mmmm...🤔🤔🤔

I think this is correct....

Thanks! :)

First, let's focus on the expression we need A and B to be equal to. 

In order to find what we need, we need to be able to compare coefficients. let's convert it to standard form. 

We have

\((x - 2)(x + 2)=x^2-4\)

However, the constans don't match. Thus, we need to divide what we have by -4 to get a constant of +1. 

We have

\(-\frac{1}{4}x^2+1\)

Now. we cam compare coefficients. Since A is the coefficient of x^2, we have

\(A=\frac{-1}{4}\)

Since B is the coefficient of x, we have \(B=0\)

Thus, we finally have 

\(A+B=-\frac{1}{4}+0 = -\frac{1}{4}\)

So -1/4 is our final answer. 

Thanks! :)

Let's create a system of equations to solve this question. 

First, we know that the two points must be on the equation \(x^2+y^2=25\)

We also know that x must be 4. 

Thus, plugging in x=4 into the first equation, we can find y values. We have

\(4^2+y^2=25\\ y^2=25-16\\ y=\pm\sqrt9\\ y=3, y=-3\)

Thus, the two points A and B are \((4, 3), (4, -3)\)

Since they share an x value, the difference between the y values is the distance. We have

\(3-(-3)=6\)

Thus, 6 is our final answer. 

Thanks! :)