Algebra Examples

$x^{4} - 6 x^{2} - 27 = 0$

Step 1

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

${(x^{2})}^{2} - 6 x^{2} - 27 = 0$

Step 2

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

$u^{2} - 6 u - 27 = 0$

Step 3

Factor $u^{2} - 6 u - 27$ 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 $- 27$ and whose sum is $- 6$ .

$- 9, 3$

Step 3.2

Write the factored form using these integers.

$(u - 9) (u + 3) = 0$

Step 4

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

$(x^{2} - 9) (x^{2} + 3) = 0$

Step 5

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

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

Step 6

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 $b = 3$ .

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

Step 7

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

$x + 3 = 0$

$x - 3 = 0$

$x^{2} + 3 = 0$

Step 8

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 8.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 9

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 9.2

Add $3$ to both sides of the equation.

$x = 3$

Step 10

Set $x^{2} + 3$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $x^{2} + 3$ equal to $0$ .

$x^{2} + 3 = 0$

Step 10.2

Solve $x^{2} + 3 = 0$ for $x$ .

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Step 10.2.1

Subtract $3$ from both sides of the equation.

$x^{2} = - 3$

Step 10.2.2

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

$x = \pm \sqrt{- 3}$

Step 10.2.3

Simplify $\pm \sqrt{- 3}$ .

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Step 10.2.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 10.2.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 10.2.3.3

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

$x = \pm i \sqrt{3}$

Step 10.2.4

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

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Step 10.2.4.1

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

$x = i \sqrt{3}$

Step 10.2.4.2

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

$x = - i \sqrt{3}$

Step 10.2.4.3

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

$x = i \sqrt{3}, - i \sqrt{3}$

Step 11

The final solution is all the values that make $(x + 3) (x - 3) (x^{2} + 3) = 0$ true.

$x = - 3, 3, i \sqrt{3}, - i \sqrt{3}$