Find the exact solutions of x^2 - (y - 12)^2 = 144 and y = -x^2.
No solution?
second verse, same as the first:
y = -x^2, so
x^2 - (-x^2 - 12)^2 = 144
x^2 - x^4 - 24x^2 - 144 = 144
x^4 + 23x^2 + 288 = 0
again, the discriminant is negative, so there are no solutions.
x=0 works only in the 2nd equation
(±12,12) only works in the first equation
Thank you!
To find the exact solutions of the given system of equations:
1. Start with the equation y = -x^2.
2. Substitute this expression for y in the first equation: x^2 - (y - 12)^2 = 144. This gives us: x^2 - (-x^2 - 12)^2 = 144.
3. Simplify the equation: x^2 - (-x^2 - 12)^2 = 144. Expanding the square term, we get: x^2 - (x^4 + 24x^2 + 144) = 144.
4. Combine like terms: x^2 - x^4 - 24x^2 - 144 = 144. Simplify further: -x^4 - 25x^2 - 288 = 0.
5. Rearrange the equation to solve for x: -x^4 - 25x^2 - 288 = 0. Notice that this is a quartic equation.
6. Unfortunately, quartic equations can be quite difficult to solve algebraically. In this case, it seems that there are no simple rational solutions for x that make the equation true.
Therefore, based on the given equations, it appears that there are no exact solutions for the system of equations.