Find the exact solutions ofx^2 - (y- 5)^2 = 25 andy= -x^2

Do I try to isolate the y in the first equation first?

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- Jul 23rd 2007, 02:07 PMBlueStarSolving Quadratic Systems
Find the exact solutions of

*x^*2 - (*y*- 5)^2 = 25 and*y*= -*x^*2

Do I try to isolate the y in the first equation first? - Jul 23rd 2007, 02:10 PMJhevon
- Jul 23rd 2007, 02:39 PMBlueStar
Would that be like x^2 - (-x^2 - 5)^2 = 25? Or should I just drop the exponent since one is over the parenthesis?

- Jul 23rd 2007, 02:51 PMJhevon
- Jul 23rd 2007, 03:09 PMBlueStar
Is the answer 0?

x^2 - (-x^2 - 5)^2 = 25

x^2 + (x^2 + 5)^2 = 25

x^2 + x^2 + 5 = 5

x^2 + x^2 = 0

2x^2 = 0

x^2 = 0

x = 0 - Jul 23rd 2007, 03:14 PMJhevon
- Jul 23rd 2007, 03:18 PMBlueStar
So would the solutions be (0, -5)?

- Jul 23rd 2007, 03:27 PMJhevon
x = 0 is NOT a solution (if we have x = 0, that would lead to -25 = 25), and if it was, since y = -x^2, if x = 0, then y = 0, so a solution would be (0,0)

I'll start you off.

$\displaystyle x^2 - \left( -x^2 - 5 \right)^2 = 25$

$\displaystyle \Rightarrow x^2 - \left( x^4 + 10x^2 + 25 \right) = 25$

$\displaystyle \Rightarrow x^2 - x^4 - 10x^2 - 25 = 25$

Now continue (it seems as though the solutions are complex, as in, not real) - Jul 23rd 2007, 03:33 PMBlueStar
http://www.mathhelpforum.com/math-he...0a18b927-1.gif

-x^4 -9x^2 = 50

??:( - Jul 23rd 2007, 03:36 PMJhevon
Note that we can rewrite this as: $\displaystyle x^4 + 9x^2 + 50 = 0$

which in turn we could rewrite as: $\displaystyle \left( x^2 \right)^2 + 9 \left( x^2 \right) + 50 = 0$

which is a quadratic eqaution in $\displaystyle x^2$, we can therefore employ the quadratic formula - Jul 23rd 2007, 03:57 PMBlueStar
http://www.mathhelpforum.com/math-he...2504fe5c-1.gifx^2 + 9x + 50 = 0

x =__-9 +- /(9)^2 - 4(1)(50)__(attempt at typing quadratic formula)

2(1)

x =__-9 +- /81 - 200__

2

But the square root number turns out to be negative... - Jul 23rd 2007, 04:08 PMJhevon
- Jul 23rd 2007, 04:14 PMBlueStar
Now I'm vaguely remembering an imaginary term,

*i*, fitting into the equation. So would it be -119*i*, or something like that? - Jul 23rd 2007, 04:27 PMJhevon
- Jul 23rd 2007, 04:35 PMBlueStar
You mean y = 9 +- 119

*i*/ -2?