Algebra Examples

$27 x^{3} + 1 = 0$

Step 1

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

${(3 x)}^{3} + 1 = 0$

Step 2

Rewrite $1$ as $1^{3}$ .

${(3 x)}^{3} + 1^{3} = 0$

Step 3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 3 x$ and $b = 1$ .

$(3 x + 1) ({(3 x)}^{2} - 3 x \cdot 1 + 1^{2}) = 0$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $3 x$ .

$(3 x + 1) (3^{2} x^{2} - 3 x \cdot 1 + 1^{2}) = 0$

Step 4.2

Raise $3$ to the power of $2$ .

$(3 x + 1) (9 x^{2} - 3 x \cdot 1 + 1^{2}) = 0$

Step 4.3

Multiply $3$ by $- 1$ .

$(3 x + 1) (9 x^{2} - 3 x \cdot 1 + 1^{2}) = 0$

Step 4.4

Multiply $- 3$ by $1$ .

$(3 x + 1) (9 x^{2} - 3 x + 1^{2}) = 0$

Step 4.5

One to any power is one.

$(3 x + 1) (9 x^{2} - 3 x + 1) = 0$

Step 5

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

$3 x + 1 = 0$

$9 x^{2} - 3 x + 1 = 0$

Step 6

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 6.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 6.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 7

Set $9 x^{2} - 3 x + 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $9 x^{2} - 3 x + 1$ equal to $0$ .

$9 x^{2} - 3 x + 1 = 0$

Step 7.2

Solve $9 x^{2} - 3 x + 1 = 0$ 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 $a = 9$ , $b = - 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (9 \cdot 1)}}{2 \cdot 9}$

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 $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 9 \cdot 1}}{2 \cdot 9}$

Step 7.2.3.1.2

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

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Step 7.2.3.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{3 \pm \sqrt{9 - 36 \cdot 1}}{2 \cdot 9}$

Step 7.2.3.1.2.2

Multiply $- 36$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 36}}{2 \cdot 9}$

Step 7.2.3.1.3

Subtract $36$ from $9$ .

$x = \frac{3 \pm \sqrt{- 27}}{2 \cdot 9}$

Step 7.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 9}$

Step 7.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.3.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 7.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 9}$

Step 7.2.3.1.7.2

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

$x = \frac{3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 9}$

Step 7.2.3.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 9}$

Step 7.2.3.1.9

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

$x = \frac{3 \pm 3 i \sqrt{3}}{2 \cdot 9}$

Step 7.2.3.2

Multiply $2$ by $9$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{18}$

Step 7.2.3.3

Simplify $\frac{3 \pm 3 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{6}$

Step 7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.2.4.1

Simplify the numerator.

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Step 7.2.4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 9 \cdot 1}}{2 \cdot 9}$

Step 7.2.4.1.2

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

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Step 7.2.4.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{3 \pm \sqrt{9 - 36 \cdot 1}}{2 \cdot 9}$

Step 7.2.4.1.2.2

Multiply $- 36$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 36}}{2 \cdot 9}$

Step 7.2.4.1.3

Subtract $36$ from $9$ .

$x = \frac{3 \pm \sqrt{- 27}}{2 \cdot 9}$

Step 7.2.4.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 9}$

Step 7.2.4.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.4.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.4.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 7.2.4.1.7.1

Factor $9$ out of $27$ .

$x = \frac{3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 9}$

Step 7.2.4.1.7.2

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

$x = \frac{3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 9}$

Step 7.2.4.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 9}$

Step 7.2.4.1.9

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

$x = \frac{3 \pm 3 i \sqrt{3}}{2 \cdot 9}$

Step 7.2.4.2

Multiply $2$ by $9$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{18}$

Step 7.2.4.3

Simplify $\frac{3 \pm 3 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{6}$

Step 7.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{6}$

Step 7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 9 \cdot 1}}{2 \cdot 9}$

Step 7.2.5.1.2

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

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Step 7.2.5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{3 \pm \sqrt{9 - 36 \cdot 1}}{2 \cdot 9}$

Step 7.2.5.1.2.2

Multiply $- 36$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 36}}{2 \cdot 9}$

Step 7.2.5.1.3

Subtract $36$ from $9$ .

$x = \frac{3 \pm \sqrt{- 27}}{2 \cdot 9}$

Step 7.2.5.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 9}$

Step 7.2.5.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.5.1.6

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

$x = \frac{3 \pm i \cdot \sqrt{27}}{2 \cdot 9}$

Step 7.2.5.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 7.2.5.1.7.1

Factor $9$ out of $27$ .

$x = \frac{3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 9}$

Step 7.2.5.1.7.2

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

$x = \frac{3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 9}$

Step 7.2.5.1.8

Pull terms out from under the radical.

$x = \frac{3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 9}$

Step 7.2.5.1.9

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

$x = \frac{3 \pm 3 i \sqrt{3}}{2 \cdot 9}$

Step 7.2.5.2

Multiply $2$ by $9$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{18}$

Step 7.2.5.3

Simplify $\frac{3 \pm 3 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{6}$

Step 7.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{6}$

Step 7.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{6}, \frac{1 - i \sqrt{3}}{6}$

Step 8

The final solution is all the values that make $(3 x + 1) (9 x^{2} - 3 x + 1) = 0$ true.

$x = - \frac{1}{3}, \frac{1 + i \sqrt{3}}{6}, \frac{1 - i \sqrt{3}}{6}$