Find All Complex Solutions 4x^4-4x^2=8

$4 x^{4} - 4 x^{2} = 8$

Step 1

Subtract $8$ from both sides of the equation.

$4 x^{4} - 4 x^{2} - 8 = 0$

Step 2

Factor the left side of the equation.

Factor $4$ out of $4 x^{4} - 4 x^{2} - 8$ .

Factor $4$ out of $4 x^{4}$ .

$4 (x^{4}) - 4 x^{2} - 8 = 0$

Factor $4$ out of $- 4 x^{2}$ .

$4 (x^{4}) + 4 (- x^{2}) - 8 = 0$

Factor $4$ out of $- 8$ .

$4 (x^{4}) + 4 (- x^{2}) + 4 (- 2) = 0$

Factor $4$ out of $4 (x^{4}) + 4 (- x^{2})$ .

$4 (x^{4} - x^{2}) + 4 (- 2) = 0$

Factor $4$ out of $4 (x^{4} - x^{2}) + 4 (- 2)$ .

$4 (x^{4} - x^{2} - 2) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$4 ({(x^{2})}^{2} - x^{2} - 2) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$4 (u^{2} - u - 2) = 0$

Factor $u^{2} - u - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$4 ((u - 2) (u + 1)) = 0$

Factor.

Replace all occurrences of $u$ with $x^{2}$ .

$4 ((x^{2} - 2) (x^{2} + 1)) = 0$

Remove unnecessary parentheses.

$4 (x^{2} - 2) (x^{2} + 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} - 2 = 0$

$x^{2} + 1 = 0$

Step 4

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Solve $x^{2} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$x^{2} = 2$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 5

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 6

The final solution is all the values that make $4 (x^{2} - 2) (x^{2} + 1) = 0$ true.

$x = \sqrt{2}, - \sqrt{2}, i, - i$

Find All Complex Solutions 4x^4-4x^2=8

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