Find the real or imaginary solutions of the equation by factoring.
x^4-5x^2=-4
To solve the equation by factoring, we can rearrange it by adding 4 to both sides:
x^4 - 5x^2 + 4 = 0
Now, notice that this equation is in the form of a quadratic equation, where x^2 represents a variable. Let's substitute a variable, say y, for x^2 to make it easier to factor:
y^2 - 5y + 4 = 0
Now, we can factor this quadratic equation:
(y - 4)(y - 1) = 0
Setting each factor equal to zero gives us:
y - 4 = 0 or y - 1 = 0
Solving these equations separately:
For y - 4 = 0: y = 4
For y - 1 = 0: y = 1
Now, substitute the variable back in:
For y = 4: x^2 = 4
Taking the square root of both sides gives us:
x = ±2
For y = 1: x^2 = 1
Taking the square root of both sides gives us:
x = ±1
Therefore, the solutions to the equation x^4 - 5x^2 = -4 are x = -2, -1, 1, and 2.