Xenosmilus

Start by moving all the terms to one side and combining like terms:

\(4x^2-8x+4y^2+14y+12=0\)

Then, divide both sides by \(4 \):

\(x^2-2x+y^2+\frac{7}{2}y+3=0\)

Next, complete the square for \(x \) and \(y \):

\((x-1)^2+(y+\frac{7}{4})^2-1-\frac{49}{16}+3=0\)

Simplify:

\((x-1)^2+(y+\frac{7}{4})^2=\frac{17}{16}\)

This is the equation (in standard form) for the circle. Now take the square root of \(\frac{17}{16} \):

\(\sqrt{\frac{17}{16}}=\frac{\sqrt{17}}{\sqrt{16}}=\boxed{\frac{\sqrt{17}}{4}}\)  Which is our answer.

The answer on that page (by CPhill) is wrong however: 

Ball splits             Ways to distribute the balls

7  1                                  2

6  2                                  2

5  3                                  2

4  4                                  1

2 * 2 * 2 * 1  =  8 ways

but you cannot multiply as these are separate cases; you add them to get 7.

Then the answer would be 75.

There is also another way (stars and bars):

Choose 3 boxes to be empty: \({5 \choose 3}=10\)

Use stars and bars allowing empty boxes to calculate (or just count like CPhill)

CPhills method if he was here:

Ball splits

0 8

1 7

2 6

3 5

4 4

5 3

6 2

7 1

8 0

to get 9 different cases (stars and bars would be \({n+k-1 \choose k-1}={8+2-1\choose 2-1}={9\choose 1}=9\) ways)

Multiply by 10 to get 90 ways. However, this overcounts (0 0 0 8 0 is counted four times, and so are all other permutations) so we subtract 5*3=15 from 90 to get our final answer, 75.