Expert: Sherman D. - 8/4/2005

Question
Find the exact solutions of x^2-(y-6)^2=36 and y=-x^2.

Answer
x^2 - (y - 6)^2 = 36
y = -x^2

x^2 - (-x^2 - 6)^2 = 36

x^2 - ((-x^2 - 6)(-x^2 - 6)) = 36

x^2 - (x^4 + 6x^2 + 6x^2 + 36) = 36

x^2 - (x^4 + 12x^2 + 36) = 36

x^2 - x^4 - 12x^2 - 36 = 36

-x^4 + (1 - 12)x^2 - 36 = 36

-x^4 - 11x^2 - 36 = 36

-x^4 - 11x^2 - 72 = 0

-(x^4 + 11x^2 + 72) = 0

x^4 + 11x^2 + 72 = 0

x = 1.22174 - 2.64436i
x = -1.22174 - 2.64436i
x = 1.22174 + 2.64436i
x = -1.22174 + 2.64436i

if you go to www.quickmath.com, you can find it for yourself.

Since this is like a quadratic equation, you can use the quadratic formula

x = (-b + sqrt(b^2 - 4ac))/2a
x = (-b - sqrt(b^2 - 4ac))/2a
whereas
a = 1
b = 11
c = 72

x = (-11 + sqrt(167)i)/2
x = (-11 - sqrt(167)i)/2
since the highest exponent is a 4 and not a 2, or its because you can have -(x^4 - 11x^2 - 72) or x^4 + 11x^2 + 72, anyway you can also say
x = (11 + sqrt(167)i)/2
x = (11 - sqrt(167)i)/2