Solve by Factoring x^4+12x^2+32=0

$x^{4} + 12 x^{2} + 32 = 0$

Step 1

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

${(x^{2})}^{2} + 12 x^{2} + 32 = 0$

Step 2

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

$u^{2} + 12 u + 32 = 0$

Step 3

Factor $u^{2} + 12 u + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Write the factored form using these integers.

$(u + 4) (u + 8) = 0$

Step 4

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

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

Step 5

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

$x^{2} + 4 = 0$

$x^{2} + 8 = 0$

Step 6

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Solve $x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

Step 7

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Solve $x^{2} + 8 = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

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

$x = \pm \sqrt{- 8}$

Simplify $\pm \sqrt{- 8}$ .

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

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

$x = \pm i \cdot \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \pm i \cdot \sqrt{4 (2)}$

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

$x = \pm i \cdot \sqrt{2^{2} \cdot 2}$

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{2})$

Move $2$ to the left of $i$ .

$x = \pm 2 i \sqrt{2}$

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.

$x = 2 i \sqrt{2}$

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

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

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

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

Step 8

The final solution is all the values that make $(x^{2} + 4) (x^{2} + 8) = 0$ true.

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

Solve by Factoring x^4+12x^2+32=0

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