Precalculus Examples

$x^{4} - 3 x^{3} = 81 + 18 x - 5 x^{3}$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $18 x$ from both sides of the equation.

$x^{4} - 3 x^{3} - 18 x = 81 - 5 x^{3}$

Step 1.2

Add $5 x^{3}$ to both sides of the equation.

$x^{4} - 3 x^{3} - 18 x + 5 x^{3} = 81$

Step 1.3

Add $- 3 x^{3}$ and $5 x^{3}$ .

$x^{4} + 2 x^{3} - 18 x = 81$

Step 2

Subtract $81$ from both sides of the equation.

$x^{4} + 2 x^{3} - 18 x - 81 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Regroup terms.

$x^{4} - 81 + 2 x^{3} - 18 x = 0$

Step 3.2

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

${(x^{2})}^{2} - 81 + 2 x^{3} - 18 x = 0$

Step 3.3

Rewrite $81$ as $9^{2}$ .

${(x^{2})}^{2} - 9^{2} + 2 x^{3} - 18 x = 0$

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$(x^{2} + 9) (x^{2} - 9) + 2 x^{3} - 18 x = 0$

Step 3.5

Simplify.

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Step 3.5.1

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

$(x^{2} + 9) (x^{2} - 3^{2}) + 2 x^{3} - 18 x = 0$

Step 3.5.2

Factor.

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Step 3.5.2.1

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 $b = 3$ .

$(x^{2} + 9) ((x + 3) (x - 3)) + 2 x^{3} - 18 x = 0$

Step 3.5.2.2

Remove unnecessary parentheses.

$(x^{2} + 9) (x + 3) (x - 3) + 2 x^{3} - 18 x = 0$

Step 3.6

Factor $2 x$ out of $2 x^{3} - 18 x$ .

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Step 3.6.1

Factor $2 x$ out of $2 x^{3}$ .

$(x^{2} + 9) (x + 3) (x - 3) + 2 x (x^{2}) - 18 x = 0$

Step 3.6.2

Factor $2 x$ out of $- 18 x$ .

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

Step 3.6.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 9)$ .

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

Step 3.7

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

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

Step 3.8

Factor.

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Step 3.8.1

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 $b = 3$ .

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

Step 3.8.2

Remove unnecessary parentheses.

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

Step 3.9

Factor $(x + 3) (x - 3)$ out of $(x^{2} + 9) (x + 3) (x - 3) + 2 x (x + 3) (x - 3)$ .

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Step 3.9.1

Factor $(x + 3) (x - 3)$ out of $(x^{2} + 9) (x + 3) (x - 3)$ .

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

Step 3.9.2

Factor $(x + 3) (x - 3)$ out of $2 x (x + 3) (x - 3)$ .

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

Step 3.9.3

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (x^{2} + 9) + (x + 3) (x - 3) (2 x)$ .

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

Step 3.10

Reorder terms.

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

Step 4

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

$x + 3 = 0$

$x - 3 = 0$

$x^{2} + 2 x + 9 = 0$

Step 5

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

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Step 5.1

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

$x + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

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

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Step 6.1

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

$x - 3 = 0$

Step 6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7

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

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Step 7.1

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

$x^{2} + 2 x + 9 = 0$

Step 7.2

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

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Step 7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

Raise $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 9}}{2 \cdot 1}$

Step 7.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 2 \pm \sqrt{4 - 36}}{2 \cdot 1}$

Step 7.2.3.1.3

Subtract $36$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 32}}{2 \cdot 1}$

Step 7.2.3.1.4

Rewrite $- 32$ as $- 1 (32)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 32}}{2 \cdot 1}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (32)}$ as $\sqrt{- 1} \cdot \sqrt{32}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{32}}{2 \cdot 1}$

Step 7.2.3.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{32}}{2 \cdot 1}$

Step 7.2.3.1.7

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 7.2.3.1.7.1

Factor $16$ out of $32$ .

$x = \frac{- 2 \pm i \cdot \sqrt{16 (2)}}{2 \cdot 1}$

Step 7.2.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (4 \sqrt{2})}{2 \cdot 1}$

Step 7.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 2 \pm 4 i \sqrt{2}}{2 \cdot 1}$

Step 7.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 i \sqrt{2}}{2}$

Step 7.2.3.3

Simplify $\frac{- 2 \pm 4 i \sqrt{2}}{2}$ .

$x = - 1 \pm 2 i \sqrt{2}$

Step 7.2.4

The final answer is the combination of both solutions.

$x = - 1 + 2 i \sqrt{2}, - 1 - 2 i \sqrt{2}$

Step 8

The final solution is all the values that make $(x + 3) (x - 3) (x^{2} + 2 x + 9) = 0$ true.

$x = - 3, 3, - 1 + 2 i \sqrt{2}, - 1 - 2 i \sqrt{2}$