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

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

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

Subtract $4$ from both sides of the equation.

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

Step 2

Subtract $2 x^{2}$ from $- x^{2}$ .

$x^{4} - 3 x^{2} - 4 = 0$

Step 3

Factor the left side of the equation.

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

${(x^{2})}^{2} - 3 x^{2} - 4 = 0$

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

$u^{2} - 3 u - 4 = 0$

Factor $u^{2} - 3 u - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

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

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

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

Rewrite $4$ as $2^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 4

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

$x + 2 = 0$

$x - 2 = 0$

$x^{2} + 1 = 0$

Step 5

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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 8

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

$x = - 2, 2, i, - i$

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

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