What are the real and complex solutions of polynomial equation?
X^4-29x^2=-100 show your work,
To find the real and complex solutions of the polynomial equation X^4 - 29X^2 = -100, we can follow the steps below:
1. Rearrange the equation to get it in the form of a quadratic equation: X^4 - 29X^2 + 100 = 0.
2. Let Y = X^2, then the equation becomes Y^2 - 29Y + 100 = 0.
3. Solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we can factor as (Y - 4)(Y - 25) = 0.
4. Now we have two possible values for Y: Y = 4 and Y = 25.
5. Substitute these values back into Y = X^2 to find the possible values for X:
For Y = 4, X^2 = 4, which gives X = ±2.
For Y = 25, X^2 = 25, which gives X = ±5.
Therefore, the real solutions are X = -5, -2, 2, 5.