Solve by Factoring x^4+6x^2+5=0

$x^{4} + 6 x^{2} + 5 = 0$

Step 1

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

${(x^{2})}^{2} + 6 x^{2} + 5 = 0$

Step 2

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

$u^{2} + 6 u + 5 = 0$

Step 3

Factor $u^{2} + 6 u + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Write the factored form using these integers.

$(u + 1) (u + 5) = 0$

Step 4

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

$(x^{2} + 1) (x^{2} + 5) = 0$

Step 5

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

$x^{2} + 1 = 0$

$x^{2} + 5 = 0$

Step 6

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 7

Set $x^{2} + 5$ equal to $0$ and solve for $x$ .

Set $x^{2} + 5$ equal to $0$ .

$x^{2} + 5 = 0$

Solve $x^{2} + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$x^{2} = - 5$

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

$x = \pm \sqrt{- 5}$

Simplify $\pm \sqrt{- 5}$ .

Rewrite $- 5$ as $- 1 (5)$ .

$x = \pm \sqrt{- 1 (5)}$

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{5}$

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

$x = \pm i \sqrt{5}$

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.

$x = i \sqrt{5}$

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

$x = - i \sqrt{5}$

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

$x = i \sqrt{5}, - i \sqrt{5}$

Step 8

The final solution is all the values that make $(x^{2} + 1) (x^{2} + 5) = 0$ true.

$x = i, - i, i \sqrt{5}, - i \sqrt{5}$

Solve by Factoring x^4+6x^2+5=0

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