Algebra Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $9 x^{3}$ from both sides of the equation.

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

Step 1.2

Subtract $9 x^{2}$ from both sides of the equation.

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

Step 2

Reorder terms.

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

Step 3

Factor out the greatest common factor from each group.

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Step 3.1

Group the first two terms and the last two terms.

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

Step 3.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

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

Step 5

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 6

Rewrite $1$ as $1^{2}$ .

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

Step 7

Factor.

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Step 7.1

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

$(- x - 1) ((3 x + 1) (3 x - 1)) = 0$

Step 7.2

Remove unnecessary parentheses.

$(- x - 1) (3 x + 1) (3 x - 1) = 0$

Step 8

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

$- x - 1 = 0$

$3 x + 1 = 0$

$3 x - 1 = 0$

Step 9

Set $- x - 1$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $- x - 1$ equal to $0$ .

$- x - 1 = 0$

Step 9.2

Solve $- x - 1 = 0$ for $x$ .

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Step 9.2.1

Add $1$ to both sides of the equation.

$- x = 1$

Step 9.2.2

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

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Step 9.2.2.1

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

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

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 9.2.2.2.2

Divide $x$ by $1$ .

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

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 10

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

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Step 10.1

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

$3 x + 1 = 0$

Step 10.2

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

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 10.2.2

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

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Step 10.2.2.1

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

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

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $3$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

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

Step 11

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

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Step 11.1

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

$3 x - 1 = 0$

Step 11.2

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

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Step 11.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 11.2.2

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

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Step 11.2.2.1

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

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

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

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

Step 11.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 12

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

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