Find the real or imaginary solutions of the equation by factoring.
x^4-5x^2=-4
To solve the equation x^4 - 5x^2 = -4 by factoring, we can rewrite it as x^4 - 5x^2 + 4 = 0.
Let's factor the left side of the equation:
(x^2 - 4)(x^2 - 1) = 0
Now we can set each factor equal to zero:
x^2 - 4 = 0 or x^2 - 1 = 0
For the first factor, we have:
(x - 2)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 2 = 0 or x + 2 = 0
Solving for x in each equation, we find:
x = 2 or x = -2
For the second factor, we have:
(x - 1)(x + 1) = 0
Setting each factor equal to zero, we get:
x - 1 = 0 or x + 1 = 0
Solving for x in each equation, we find:
x = 1 or x = -1
Therefore, the solutions to the equation x^4 - 5x^2 = -4 are x = 2, x = -2, x = 1, and x = -1.
To find the solutions of the equation x^4 - 5x^2 = -4 by factoring, we can rearrange the equation to have zero on one side:
x^4 - 5x^2 + 4 = 0
Now, we can factor this quadratic equation. Since all the terms have even powers of x, we can substitute a variable to simplify the equation:
Let y = x^2
The equation becomes:
y^2 - 5y + 4 = 0
We can now factor this quadratic equation:
(y - 4)(y - 1) = 0
Setting each factor equal to zero:
y - 4 = 0 or y - 1 = 0
Solving for y:
y = 4 or y = 1
Now, substituting back in terms of x:
x^2 = 4 or x^2 = 1
Taking the square root of both sides:
x = ±2 or x = ±1
Therefore, the solutions to the equation x^4 - 5x^2 = -4 are x = ±2 or x = ±1.