Basic Math Examples

$18 y^{3} = 50 y$

Step 1

Subtract $50 y$ from both sides of the equation.

$18 y^{3} - 50 y = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2 y$ out of $18 y^{3} - 50 y$ .

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Step 2.1.1

Factor $2 y$ out of $18 y^{3}$ .

$2 y (9 y^{2}) - 50 y = 0$

Step 2.1.2

Factor $2 y$ out of $- 50 y$ .

$2 y (9 y^{2}) + 2 y (- 25) = 0$

Step 2.1.3

Factor $2 y$ out of $2 y (9 y^{2}) + 2 y (- 25)$ .

$2 y (9 y^{2} - 25) = 0$

Step 2.2

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

$2 y ({(3 y)}^{2} - 25) = 0$

Step 2.3

Rewrite $25$ as $5^{2}$ .

$2 y ({(3 y)}^{2} - 5^{2}) = 0$

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 $a = 3 y$ and $b = 5$ .

$2 y ((3 y + 5) (3 y - 5)) = 0$

Step 2.4.2

Remove unnecessary parentheses.

$2 y (3 y + 5) (3 y - 5) = 0$

Step 3

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

$y = 0$

$3 y + 5 = 0$

$3 y - 5 = 0$

Step 4

Set $y$ equal to $0$ .

$y = 0$

Step 5

Set $3 y + 5$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $3 y + 5$ equal to $0$ .

$3 y + 5 = 0$

Step 5.2

Solve $3 y + 5 = 0$ for $y$ .

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Step 5.2.1

Subtract $5$ from both sides of the equation.

$3 y = - 5$

Step 5.2.2

Divide each term in $3 y = - 5$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 y = - 5$ by $3$ .

$\frac{3 y}{3} = \frac{- 5}{3}$

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.

$\frac{3 y}{3} = \frac{- 5}{3}$

Step 5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{3}$

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.

$y = - \frac{5}{3}$

Step 6

Set $3 y - 5$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $3 y - 5$ equal to $0$ .

$3 y - 5 = 0$

Step 6.2

Solve $3 y - 5 = 0$ for $y$ .

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Step 6.2.1

Add $5$ to both sides of the equation.

$3 y = 5$

Step 6.2.2

Divide each term in $3 y = 5$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 y = 5$ by $3$ .

$\frac{3 y}{3} = \frac{5}{3}$

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.

$\frac{3 y}{3} = \frac{5}{3}$

Step 6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{3}$

Step 7

The final solution is all the values that make $2 y (3 y + 5) (3 y - 5) = 0$ true.

$y = 0, - \frac{5}{3}, \frac{5}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = 0, - \frac{5}{3}, \frac{5}{3}$

Decimal Form:

$y = 0, - 1.\overline{6}, 1.\overline{6}$

Mixed Number Form:

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