Algebra Examples

$x^{6} + x^{3} - 12 = 0$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} + x^{3} - 12 = 0$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

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

Step 3

Factor $u^{2} + u - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 3.2

Write the factored form using these integers.

$(u - 3) (u + 4) = 0$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

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

Step 5

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

$x^{3} - 3 = 0$

$x^{3} + 4 = 0$

Step 6

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

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Step 6.1

Set $x^{3} - 3$ equal to $0$ .

$x^{3} - 3 = 0$

Step 6.2

Solve $x^{3} - 3 = 0$ for $x$ .

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Step 6.2.1

Add $3$ to both sides of the equation.

$x^{3} = 3$

Step 6.2.2

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

$x = \sqrt[3]{3}$

Step 7

Set $x^{3} + 4$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{3} + 4$ equal to $0$ .

$x^{3} + 4 = 0$

Step 7.2

Solve $x^{3} + 4 = 0$ for $x$ .

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Step 7.2.1

Subtract $4$ from both sides of the equation.

$x^{3} = - 4$

Step 7.2.2

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

$x = \sqrt[3]{- 4}$

Step 7.2.3

Simplify $\sqrt[3]{- 4}$ .

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Step 7.2.3.1

Rewrite $- 4$ as ${(- 1)}^{3} \cdot 4$ .

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Step 7.2.3.1.1

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

$x = \sqrt[3]{- 1 (4)}$

Step 7.2.3.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 4}$

Step 7.2.3.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{4}$

Step 7.2.3.3

Rewrite $- 1 \sqrt[3]{4}$ as $- \sqrt[3]{4}$ .

$x = - \sqrt[3]{4}$

Step 8

The final solution is all the values that make $(x^{3} - 3) (x^{3} + 4) = 0$ true.

$x = \sqrt[3]{3}, - \sqrt[3]{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[3]{3}, - \sqrt[3]{4}$

Decimal Form:

$x = 1.44224957 \dots, - 1.58740105 \dots$