SOLUTION: Find the points of intersection, if any, of the graphs in the system. Give the solution of: 4x^2 - 56x + 9y^2 + 160 = 0 and 4x^2 + y^2 - 64 = 0. Please and thank you. please please

Algebra ->  Systems-of-equations -> SOLUTION: Find the points of intersection, if any, of the graphs in the system. Give the solution of: 4x^2 - 56x + 9y^2 + 160 = 0 and 4x^2 + y^2 - 64 = 0. Please and thank you. please please      Log On


   

Question 288664: Find the points of intersection, if any, of the graphs in the system. Give the solution of: 4x^2 - 56x + 9y^2 + 160 = 0 and 4x^2 + y^2 - 64 = 0. Please and thank you. please please help me find the right solution for x and y, I had some help and they told me that the equation to solve was 4x^2-56x+9-(-4x^2+64)+160=0, but I'm stuck on how to solve this, please help! thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4x%5E2+%2B+y%5E2+-+64+=+0 Start with the second equation.


y%5E2+=+-4x%5E2%2B64 Solve for y%5E2 by getting every other term to the right side.


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4x%5E2+-+56x+%2B+9y%5E2+%2B+160+=+0 Move back to the first equation.


4x%5E2+-+56x+%2B+9%28-4x%5E2%2B64%29+%2B+160+=+0 Replace each y%5E2 term with +-4x%5E2%2B64 (since the two are essentially the same or equivalent).


4x%5E2+-+56x+-36x%5E2%2B576+%2B+160+=+0 Distribute.


-32x%5E2+-+56x%2B736+=+0 Combine like terms.


Now let's solve -32x%5E2+-+56x%2B736+=+0 by use of the quadratic formula.


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve -32%2Ax%5E2-56%2Ax%2B736=0 ( notice a=-32, b=-56, and c=736)





x+=+%28--56+%2B-+sqrt%28+%28-56%29%5E2-4%2A-32%2A736+%29%29%2F%282%2A-32%29 Plug in a=-32, b=-56, and c=736




x+=+%2856+%2B-+sqrt%28+%28-56%29%5E2-4%2A-32%2A736+%29%29%2F%282%2A-32%29 Negate -56 to get 56




x+=+%2856+%2B-+sqrt%28+3136-4%2A-32%2A736+%29%29%2F%282%2A-32%29 Square -56 to get 3136 (note: remember when you square -56, you must square the negative as well. This is because %28-56%29%5E2=-56%2A-56=3136.)




x+=+%2856+%2B-+sqrt%28+3136%2B94208+%29%29%2F%282%2A-32%29 Multiply -4%2A736%2A-32 to get 94208




x+=+%2856+%2B-+sqrt%28+97344+%29%29%2F%282%2A-32%29 Combine like terms in the radicand (everything under the square root)




x+=+%2856+%2B-+312%29%2F%282%2A-32%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%2856+%2B-+312%29%2F-64 Multiply 2 and -32 to get -64


So now the expression breaks down into two parts


x+=+%2856+%2B+312%29%2F-64 or x+=+%2856+-+312%29%2F-64


Lets look at the first part:


x=%2856+%2B+312%29%2F-64


x=368%2F-64 Add the terms in the numerator

x=-23%2F4 Divide


So one answer is

x=-23%2F4




Now lets look at the second part:


x=%2856+-+312%29%2F-64


x=-256%2F-64 Subtract the terms in the numerator

x=4 Divide


So another answer is

x=4


So our solutions are:

x=-23%2F4 or x=4





Since the solutions in terms of 'x' are x=-23%2F4 or x=4, we can use them to find the corresponding solutions in terms of 'y'.


So if x=-23%2F4, then...


y%5E2+=+-4x%5E2%2B64 Start with the given equation


y%5E2+=+-4%28-23%2F4%29%5E2%2B64 Plug in x=-23%2F4


y%5E2+=+-4%28529%2F16%29%2B64 Square -23%2F4 to get -529%2F16


y%5E2+=+-2116%2F16%2B64 Multiply


y%5E2+=+-529%2F4%2B64 Reduce.


y%5E2+=+-273%2F4 Combine like terms.


y+=+%22%22%2B-sqrt%28-273%2F4%29 Take the square root of both sides.


Since the square root of a negative number is not a real number, this means that there are no real solutions of 'y' when x=-23%2F4. So we can ignore this value.


Now if x=4, then...


y%5E2+=+-4x%5E2%2B64 Start with the given equation


y%5E2+=+-4%284%29%5E2%2B64 Plug in x=4


y%5E2+=+-4%2816%29%2B64 Square 4 to get 16.


y%5E2+=+-64%2B64 Multiply


y%5E2+=+0 Combine like terms.


y=sqrt%280%29 Take the square root of both sides.


y=0 Take the square root of 0 to get 0.


So when x=4, y=0 giving us the ordered pair (4,0)


This means that the two graphs intersect at the only point of (4,0)