Basic Math Examples

$12 y^{3} = 3 y$

Step 1

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

$12 y^{3} - 3 y = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $3 y$ out of $12 y^{3} - 3 y$ .

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Step 2.1.1

Factor $3 y$ out of $12 y^{3}$ .

$3 y (4 y^{2}) - 3 y = 0$

Step 2.1.2

Factor $3 y$ out of $- 3 y$ .

$3 y (4 y^{2}) + 3 y (- 1) = 0$

Step 2.1.3

Factor $3 y$ out of $3 y (4 y^{2}) + 3 y (- 1)$ .

$3 y (4 y^{2} - 1) = 0$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Factor.

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Step 2.4.1

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

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

Step 2.4.2

Remove unnecessary parentheses.

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

Step 3

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

$y = 0$

$2 y + 1 = 0$

$2 y - 1 = 0$

Step 4

Set $y$ equal to $0$ .

$y = 0$

Step 5

Set $2 y + 1$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $2 y + 1$ equal to $0$ .

$2 y + 1 = 0$

Step 5.2

Solve $2 y + 1 = 0$ for $y$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$2 y = - 1$

Step 5.2.2

Divide each term in $2 y = - 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 y = - 1$ by $2$ .

$\frac{2 y}{2} = \frac{- 1}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 1}{2}$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 1}{2}$

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.

$y = - \frac{1}{2}$

Step 6

Set $2 y - 1$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $2 y - 1$ equal to $0$ .

$2 y - 1 = 0$

Step 6.2

Solve $2 y - 1 = 0$ for $y$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$2 y = 1$

Step 6.2.2

Divide each term in $2 y = 1$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 y = 1$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2}$

Step 6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2}$

Step 7

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

$y = 0, - \frac{1}{2}, \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = 0, - \frac{1}{2}, \frac{1}{2}$

Decimal Form:

$y = 0, - 0.5, 0.5$