SOLUTION: What is the points of intersection of these equations x-2y-8=0 x^2=y^2-12x+6y+29=0

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Question 258000: What is the points of intersection of these equations
x-2y-8=0
x^2=y^2-12x+6y+29=0
Found 3 solutions by Greenfinch, Edwin McCravy, Alan3354:
Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
Second equation is a circle, so reorganise it
(x^2 + 12 x) - (y^2 - 6y)+ 29 = 0 and then complete the square
(x^2 + 12x + 36) - ( y^2 - 6y + 9) - 45 + 29 = 0 which becomes
(x + 6)^2 - (y - 3)^2 = 4^2 which is a circle centre at -6, 3 and radius 4
other equation is y = (1/2)x - 4 or x = 2y + 8
Probably solve for y is easier
(2y + 14)^2 - (y - 3)^2 = 4^2
4y^2 + 56 y + 196 -(y^2 - 6y +9) = 16
3y^2 + 62y + 187 = 16
3y^2 + 62y + 171 = 0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case 3y%5E2%2B62y%2B171+=+0) has the following solutons:

y%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2862%29%5E2-4%2A3%2A171=1792.

Discriminant d=1792 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-62%2B-sqrt%28+1792+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%2862%29%2Bsqrt%28+1792+%29%29%2F2%5C3+=+-3.27799650382776
y%5B2%5D+=+%28-%2862%29-sqrt%28+1792+%29%29%2F2%5C3+=+-17.3886701628389

Quadratic expression 3y%5E2%2B62y%2B171 can be factored:
3y%5E2%2B62y%2B171+=+3%28y--3.27799650382776%29%2A%28y--17.3886701628389%29
Again, the answer is: -3.27799650382776, -17.3886701628389. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B62%2Ax%2B171+%29

Then use x = 2y + 8 to get the x figures

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!


Note: The other two tutors' solutions are both incorrect.


system%28x-2y-8=0%2C+x%5E2%2By%5E2-12x%2B6y%2B29=0%29

Solve the first equation for x

x-2y-8=0
x=2y%2B8

Substitute %282y%2B8%29 for x in the second equation:

x%5E2%2By%5E2-12x%2B6y%2B29=0

%282y%2B8%29%5E2%2By%5E2-12%282y%2B8%29%2B6y%2B29=0

%282y%2B8%29%282y%2B8%29%2By%5E2-12%282y%2B8%29%2B6y%2B29=0

%284y%5E2%2B16y%2B16y%2B64%29%2By%5E2-12%282y%2B8%29%2B6y%2B29=0

4y%5E2%2B32y%2B64%2By%5E2-24y-96%2B6y%2B29=0

5y%5E2%2B14y-3=0

Factor:

%285y-1%29%28y%2B3%29=0

Use zero-factor principle:

5y-1=0 gives solution y=1%2F5

y%2B3=0 gives solution y=-3

Now we must find the value of x for each of these
two values for y.

For y=1%2F5 we substitute %281%2F5%29 for y in 

x=2y%2B8

x=2%281%2F5%29%2B8

x=2%2F5%2B8

x=2%2F5%2B40%2F5

x=42%2F5%29+%0D%0A%0D%0ASo+one+solution+is+%28x%2Cy%29+=+%28%7B%7B%7B42%2F5,1%2F5)

or if you prefer, (x,y) = (8.4,.2)

For y=-3 we substitute %28-3%29 for y in 

x=2y%2B8

x=2%28-3%29%2B8

x=-6%2B8

x=2

So the other solution is (x,y) = (2,-3)

Edwin

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The 2nd "equation" has too many equal signs.
If the 1st = is a +, then Edwin's solution is good.
If it's a minus sign, it's a different solution.
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You can see in Greenfinch's photo that he's sound asleep.
Greenfinch said it's a circle, but used a minus sign which makes it a hyperbola.