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

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Factor by grouping

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Q. Factorize.

x^4-x^2-20=0

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.

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