What are the real and complex solutions of the polynomial equation?
x^4-29x^2=-100
show all steps/work pls
To solve the given equation x^4 - 29x^2 = -100, we can rearrange it as follows:
x^4 - 29x^2 + 100 = 0.
Now, let's substitute a variable, such as y, to make the equation easier to solve:
y = x^2.
Substituting y into the equation, we get:
y^2 - 29y + 100 = 0.
Now we have a quadratic equation in terms of y. We can solve this equation by factoring or using the quadratic formula. In this case, we will factor:
(y - 4)(y - 25) = 0.
Setting each factor equal to zero, we get:
y - 4 = 0 ---> y = 4,
y - 25 = 0 ---> y = 25.
Now, substitute y back into the equation y = x^2:
x^2 = 4 ---> x = ±√4 ---> x = ±2,
x^2 = 25 ---> x = ±√25 ---> x = ±5.
Thus, the real solutions of the equation x^4 - 29x^2 = -100 are x = 2 and x = 5.
Now let's find the complex solutions:
Since x^2 = 4 has already been solved as x = ±2, we can ignore this equation.
For x^2 = 25:
x = ±√25 ---> x = ±5.
Therefore, the complex solutions are x = 5i and x = -5i.
Therefore, the solutions to the polynomial equation x^4 - 29x^2 = -100 are:
Real solutions: x = 2 and x = 5.
Complex solutions: x = 5i and x = -5i.