What are the real and complex solutions of polynomial equation?
X^4-29x^2=-100 show your work, explain without words use numbers
To find the real and complex solutions of the polynomial equation x^4 - 29x^2 + 100 = 0, we can use a quadratic substitution method to simplify the equation.
Let's introduce a new variable, u = x^2. We then have u^2 - 29u + 100 = 0.
Now, we can solve this quadratic equation for u by factoring or using the quadratic formula. Factoring, we have:
(u - 25)(u - 4) = 0
Setting each factor equal to zero:
u - 25 = 0 or u - 4 = 0
Solving for u:
u = 25 or u = 4
Substituting back for u = x^2:
x^2 = 25 or x^2 = 4
Taking the square root of both sides:
x = ±√25 or x = ±√4
Therefore, the real solutions are x = ±5, x = ±2. These are the values of x that satisfy the given polynomial equation.
Since the square root of a negative number does not exist in the realm of real numbers, there are no complex solutions for this equation.