Finite Math Examples

$9 x^{4} + 18 x^{3} - x^{2} - 2 x = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x$ out of $9 x^{4} + 18 x^{3} - x^{2} - 2 x$ .

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Step 1.1.1

Factor $x$ out of $9 x^{4}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.1.5

Factor $x$ out of $x (9 x^{3}) + x (18 x^{2})$ .

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

Step 1.1.6

Factor $x$ out of $x (9 x^{3} + 18 x^{2}) + x (- x)$ .

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

Step 1.1.7

Factor $x$ out of $x (9 x^{3} + 18 x^{2} - x) + x \cdot - 2$ .

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

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

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Factor.

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Step 1.6.1

Factor.

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Step 1.6.1.1

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 $b = 1$ .

$x ((x + 2) ((3 x + 1) (3 x - 1))) = 0$

Step 1.6.1.2

Remove unnecessary parentheses.

$x ((x + 2) (3 x + 1) (3 x - 1)) = 0$

Step 1.6.2

Remove unnecessary parentheses.

$x (x + 2) (3 x + 1) (3 x - 1) = 0$

Step 2

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

$x = 0$

$x + 2 = 0$

$3 x + 1 = 0$

$3 x - 1 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

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Step 5.1

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

$3 x + 1 = 0$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 5.2.2

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

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Step 5.2.2.1

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

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Move the negative in front of the fraction.

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

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

Add $1$ to 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 7

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

$x = 0, - 2, - \frac{1}{3}, \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - 2, - \frac{1}{3}, \frac{1}{3}$

Decimal Form:

$x = 0, - 2, - 0.\overline{3}, 0.\overline{3}$