Basic Math Examples

$3 a - 2 = 27 a^{3} - 18 a^{2}$

Step 1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$27 a^{3} - 18 a^{2} = 3 a - 2$

Step 2

Subtract $3 a$ from both sides of the equation.

$27 a^{3} - 18 a^{2} - 3 a = - 2$

Step 3

Add $2$ to both sides of the equation.

$27 a^{3} - 18 a^{2} - 3 a + 2 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor out the greatest common factor from each group.

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Step 4.1.1

Group the first two terms and the last two terms.

$(27 a^{3} - 18 a^{2}) - 3 a + 2 = 0$

Step 4.1.2

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

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

Step 4.2

Factor the polynomial by factoring out the greatest common factor, $3 a - 2$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Factor.

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Step 4.5.1

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

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

Step 4.5.2

Remove unnecessary parentheses.

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

Step 5

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

$3 a - 2 = 0$

$3 a + 1 = 0$

$3 a - 1 = 0$

Step 6

Set $3 a - 2$ equal to $0$ and solve for $a$ .

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Step 6.1

Set $3 a - 2$ equal to $0$ .

$3 a - 2 = 0$

Step 6.2

Solve $3 a - 2 = 0$ for $a$ .

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Step 6.2.1

Add $2$ to both sides of the equation.

$3 a = 2$

Step 6.2.2

Divide each term in $3 a = 2$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 a = 2$ by $3$ .

$\frac{3 a}{3} = \frac{2}{3}$

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.

$\frac{3 a}{3} = \frac{2}{3}$

Step 6.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{3}$

Step 7

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

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Step 7.1

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

$3 a + 1 = 0$

Step 7.2

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

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Step 7.2.1

Subtract $1$ from both sides of the equation.

$3 a = - 1$

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

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

Step 8

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

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Step 8.1

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

$3 a - 1 = 0$

Step 8.2

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

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Step 8.2.1

Add $1$ to both sides of the equation.

$3 a = 1$

Step 8.2.2

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

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Step 8.2.2.1

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

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

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.2.1.2

Divide $a$ by $1$ .

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

Step 9

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

$a = \frac{2}{3}, - \frac{1}{3}, \frac{1}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$a = \frac{2}{3}, - \frac{1}{3}, \frac{1}{3}$

Decimal Form:

$a = 0.\overline{6}, - 0.\overline{3}, 0.\overline{3}$