Factorize. x^4-x^2-20=0

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Recognize Quadratic Form: Let's first recognize that this is a quadratic in form, where x^2 is the variable. We can substitute x^2 with a new variable, let's say u, to make it look like a standard quadratic equation. So, let u = x^2. The equation becomes u^2 - u - 20 = 0.Substitute Variable u: Now, we factor the quadratic equation u^2 - u - 20. We are looking for two numbers that multiply to -20 and add up to -1. These numbers are -5 and 4. So we can write the equation as (u - 5)(u + 4) = 0.Factor Quadratic Equation: Next, we substitute back x^2 for u to get the factors in terms of x. This gives us (x^2 - 5)(x^2 + 4) = 0.Substitute back to x: Now we need to solve for x. We have two separate factors that can be set to zero: x^2 - 5 = 0 and x^2 + 4 = 0. We will solve each one separately.Solve for x - Factor 1: For the first factor, x^2 - 5 = 0, we add 5 to both sides to get x^2 = 5. Taking the square root of both sides gives us two solutions: x = sqrt(5) and x = -sqrt(5).Solve for x - Factor 2: For the second factor, x^2 + 4 = 0, we subtract 4 from both sides to get x^2 = -4. Taking the square root of both sides gives us two complex solutions: x = 2i and x = -2i, where i is the imaginary unit.

Factorize. $x^4-x^2-20=0$

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Factorize. $x^4-x^2-20=0$