NotThatSmart

First, let's simplify the equation. We have

\(x^2 - 6x + 13\)

Since the constant is equal to uv, we have the equation

\(uv = 13 \\ 2uv = 26\)

The coefficient of x is equal to u+v, so we have

\(u + v = 6\)

Squaring both sides of the equation, we have the equation

\(u^2 + 2uv + v^2 = 36 \\ u^2 + 26 + v^2 = 36 \\ u^2 + v^2 = 10\)

Now, it's time for the fun part of this problem. We can simplify what we want to figure out to

We already know all these terms! We can easily find them! We have

\(u^3 + v^3 = ( u + v) ( u^2 + v^2 - uv) = (6)(10 - 13) = -18 \\ [uv]^2 = [ 13]^2 = 169\)

So, The answer is \(u/v^2 + v/u^2 = [ u^3 + v^3 ] / [ uv]^2 = -18 / 169\)

Thanks! :)

First of all, let's move almost all the variables to one side of the equation and combine all the like terms. 

We get \(x^2 - 21x + y^2 + 13y = 25\). 

Now, we complete the square for both x and y on the right side of the equation. 

\(x^2 - 21x + 441/4 + y^2 + 13y + 169/4 = 25 + 441/4 + 169/4\)

\((x - 21/2)^2 + (y + 13/2)^2 = 355 / 2\)

Wait! This looks like the equation for a circle! 

Indeed, we have a circle at center  \((21/2 , -13/2)\) with radius \(\sqrt{355 / 2}\). 

The value of the largest possible x is just the radius added onto the x value of the center. 

\(x = (21/2) + \sqrt{355 / 2}\)

Thanks! :)

We can use variables to complete this problem.

Let's let P = (x,y)

Plugging this value of P into what we know, we have

\(PA^2 + PB^2 + PC^2   = \\ (x-7)^2 + (y + 11)^2  + (x-10)^2 + (y-13)^2 + (x-18)^2 + ( y + 22)^2\)

\(3 x^2 - 70 x + 3 y^2 + 40 y + 1247 \)
\(3 [ x^2 - 70/3 x + y^2 + 40/3 y ] + 1247 \)

Now, we must complete the square for x and y. Doing this, we get

\(3[ x^2 - 70/3 x + 4900/36 + y^2 + 40/3y + 1600/36 ] + 1247 - 4900/12 -1600/12 \\ 3 [ (x - 70/6]^2 + (y + 40/6)^2 ] +  2116 / 3 \\ Q = (70/6, -40/6 ) =  (35/3 , - 20/3)\)

Thus, we have k = 2116 / 3

So our final answer is 2116/3. 

Thanks! :)

First, we know the equation of a sliope is \(\frac{y_2-y_1}{x_2-x_1}\)

In this problem, we know the two points we have are (a, a^2) and (b, b^2). 

Plugging these two points in, we get

\( [ b^2 - a^2 ] / [ b - a ]\)

Using the difference of squares equation for the numerator, we have

\( [ (b -a) (b + a) ] / (b -a)\)

Notice the b - a cancel each other out. 

This leaves us with

\( b + a \)

Since the slope is 2, we simplfy have

\( b + a = 2\)

So the answer is 2. 

Thanks! :)

Let's set some variables to solve this question. 

Now, since AD bisects  BAC

Let's let BD  = 3 - x

Let's let  CD =  x

Now, we have the formula

\( BD / AB = CD / AC \\ (3 - x) / 4 = x / 5\\ 5 (3 - x) = 4x \\ 15 - 5x = 4x \\ 15 = 9x \\ x = 15/9 = 5/3 = CD\)

Thus, we finally can caclulate ADC as

\([ ADC ] = (1/2) (CD) ( AB) = (1/2) ( 5/3) ( 4) = 10 / 3 \)

Thus, The answer is \(10/3\)

Thanks! :)

Let's make some obersvations. We have that
\(L =   \sqrt {92^2  + (33-25)^2} =   \sqrt{ 92^2 + 8^2} =  4\sqrt {533} \\ R = 33 \\ r =25\)

Surface area = \(\pi L (R + r) + \pi (R^2 + r^2)  = \pi [ 4\sqrt{533} ](58) + \pi ( 33^2 + 25^2)  ≈ 22211.49 units^2 \)

\(S1  =  \pi * 33^2    \\       S2  = \pi * 25^2    \\   H  = 92 \)

Volume  = 

\( H/3 * ( S1 + S2 +  \sqrt  { S1 * S2 })  =  \\ 92/3  *  ( \pi (33^2 + 25^2) +  \sqrt{\pi*33^2 * \pi* 25^2} ) ≈ 244612.78 units^3\)

Thus, The answer is \( 244612.78\)

Thanks! :)

First, let's find the radius of the circle. We simply have

\(25 \pi = \pi *r^2 \\ 25 = r^2 \\r = 5 \)

Let's let R be the radius of the sphere. 

\(R^2 = [ 5^2 + r^2 ] = [ 5^2 + 5^2 ] = 50 \)

Now, we can calculate the surface area of the sphere. We have

\(4 \pi R^2 = \\ 4 \pi * 50 = \\ 200 \pi\)

So, the answer is \(200 \pi \)

Thanks! :)

WAIT a second...

"The length of $\overline{AB}$ is $4,$ and the circle has a radius of $5.$  Find the length $AB.$"

I think we know the answer is 4 now 😁

The length of AB is \(4\)

Thanks! :)

Let's set up variables to complete this problem. 

Since AD bisects  BAC

Lets let  BD  = 3 - x

Lets let  CD =  x

Now, we can easily solve the problem with an equation. We have

\(BD / AB = CD / AC\\ (3 - x) / 4 = x / 5\\ 5 (3 - x) = 4x\\ 15 - 5x = 4x \\ 15 = 9x \\ x = 15/9 = 5/3 = CD\)

Thus, taking this value of CD and plugging it into the area equation, we find

\([ ADC ] = (1/2) (CD) ( AB) = (1/2) ( 5/3) ( 4) = 10 / 3 \)

thus, The answer is \(10/3\)

Thanks! :)

The answer is \(48\)

The factors of 48 are \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)

There are exactly 10. 

Thanks! :)