NotThatSmart

If we drew the graph for all the points 5 units away from (2, 7), we would get a circle with radius \(\sqrt5\) and center (2, 7).

A circle like that would have the equation \((x - 2)^2 + (y -7)^2 = 25\). 

Now, we just have to find the intersection points with the graph of \(y = 5x - 28\). 

We get a system of equations like this, and we just have to find (x, y). Luckily, y is already isolated in the second equation, so we just have to sub that into the first equation. We get

\((x -2)^2 + (5x -28 - 7)^2 = 25 \\ (x - 2)^2 + (5x - 35)^2 = 25\)

\(x^2 - 4x + 4 + 25x^2 - 350x + 1225 = 25 \\ 26x^2 - 354x + 1204 = 0\)

From this, we get\(x=86/13, 7\)

Plugging these back for y, we get \(y = 66/13, 7\). 

This means our two points are \((86/13, 66/13) & (7, 7)\)

Thanks! :) 

We can easily just graph out the numbers since there are not a lot of terms we have to go through.  

This means when Laverne says 12, Shirley goes off running!

Thanks! :)

Not too bad of a problem!

Let's let n be the constant we add to the 3 numbers. Lets try to figure out the 3 new numbers first. 

Since all terms form a geometric sequence, we can write

\(\frac{120+n}{60+n}=\frac{160+n}{120+n}\\ (120+n)^2=(60+n)(160+n)\\ 14400+n^2+240n=9600+n^2+220n\\ 4800=-20n\\ n=-240\)

Now, I know the n is negative, when if we apply it, we dot get -180, -120, and -80, which do form a geometric series. 

-180/-120 = 3/2, so 3/2 is the answer. 

I'm not sure if this is the correct answer, but it does work. 

Thanks! :) 

I'm going to do this the hard way and expand eveyrhing out. 

Expanding the awful polynomial, which I DEFINTELY NOT reccomending, we have

\(x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14\). 

As you can seem the coefficient is just -9. 

If you want to simplify the process, just note this. 

In the expansion, the only two terms that matters is 1*-23x amd x*14x. Adding these 2 we get -9. 

Thanks! :) 

To complete this problem, we must make an important observation.

NOTE: 

\(3 + 4 + 5 = 3 *4 \\ 13 + 14 + 15 = 3 * 14\)

Now, we just have to figure out \(3 [ 4 + 14 + 24 + ....... + 994 ] = \)

There are 100 terms in the sequence, so we have

\(3 [ 994 + 4 ] [ 100 / 2 ] = 149,700\). 

Thanks! :)

First, let's multiply it out. \(x^3(x^2+tx) = x^5+tx^4\). 

As you can see, there is no x^2 term in this expansion. 

This means t can be pretty much anything we want as long as it doesn't contain x^-2 in it!

Thanks! :)

First off, let's move all the terms to one side of the equation before combining like terms!

\(x^2 - 4x + y^2 - 2y = 0 \)

Completing the square for x and y, we get

\(x^2 - 4x + 4 + y^2 - 2y + 1 = 4 +1 \\ (x- 2)^2 + ( y - 1)^2 = 5\)

Wait! This is the equation for a circle. 

Using the equations for circles, this circle has a radius of \(\sqrt{5}\) and center (2, 1). 

The largest value of x would be \(x = 2 +\sqrt 5\).

The reason for this is because the largest value of x would just be the radius plus the x value of the center!

Thanks! :)

First off, let's move all terms to one side and combine all like terms. 

We get \(2x^2+20x+18=0\). Simplifying, we get \(x^2+10x+9 = 0\). 

Now, let's note that this quadratic can be factored! Putting it into factored form, we get \((x+1)(x+9)=0\). 

This gets us the values of \(x = -1, -9\). 

Check your answers and make sure they work. These do if we plug them back in!

Thanks! :)

After Dirk finally finishes the race, Edith has ran 14 km while Foley has ran 9 km. 

This essentially means that Foley is 9/14 as fast as Edith. 

This means when Edith has ran 24 km, Foley has ran \(24 (9/14) ≈ 15.43 \) km. 

So Foley is \(24 - 15.43 ≈ 8.57\)km behind Edith!

Thanks! :)