Algebra Examples

$x^{4} + 13 x^{2} + 36 = 0$

Step 1

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

${(x^{2})}^{2} + 13 x^{2} + 36 = 0$

Step 2

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

$u^{2} + 13 u + 36 = 0$

Step 3

Factor $u^{2} + 13 u + 36$ using the AC method.

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Step 3.1

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 $36$ and whose sum is $13$ .

$4, 9$

Step 3.2

Write the factored form using these integers.

$(u + 4) (u + 9) = 0$

Step 4

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

$(x^{2} + 4) (x^{2} + 9) = 0$

Step 5

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

$x^{2} + 4 = 0$

$x^{2} + 9 = 0$

Step 6

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

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Step 6.1

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

$x^{2} + 4 = 0$

Step 6.2

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

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Step 6.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 6.2.2

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

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.3.4

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

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

Step 6.2.3.5

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

$x = \pm i \cdot 2$

Step 6.2.3.6

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

$x = \pm 2 i$

Step 6.2.4

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

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Step 6.2.4.1

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

$x = 2 i$

Step 6.2.4.2

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

$x = - 2 i$

Step 6.2.4.3

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

$x = 2 i, - 2 i$

Step 7

Set $x^{2} + 9$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 7.2

Solve $x^{2} + 9 = 0$ for $x$ .

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Step 7.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 7.2.2

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

$x = \pm \sqrt{- 9}$

Step 7.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 7.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 7.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 7.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 7.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 7.2.3.5

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

$x = \pm i \cdot 3$

Step 7.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 7.2.4

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

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Step 7.2.4.1

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

$x = 3 i$

Step 7.2.4.2

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

$x = - 3 i$

Step 7.2.4.3

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

$x = 3 i, - 3 i$

Step 8

The final solution is all the values that make $(x^{2} + 4) (x^{2} + 9) = 0$ true.

$x = 2 i, - 2 i, 3 i, - 3 i$