Finite Math Examples

$y^{3} - 27 y = 0$

Step 1

Factor $y$ out of $y^{3} - 27 y$ .

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Step 1.1

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

$y \cdot y^{2} - 27 y = 0$

Step 1.2

Factor $y$ out of $- 27 y$ .

$y \cdot y^{2} + y \cdot - 27 = 0$

Step 1.3

Factor $y$ out of $y \cdot y^{2} + y \cdot - 27$ .

$y (y^{2} - 27) = 0$

Step 2

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

$y = 0$

$y^{2} - 27 = 0$

Step 3

Set $y$ equal to $0$ .

$y = 0$

Step 4

Set $y^{2} - 27$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $y^{2} - 27$ equal to $0$ .

$y^{2} - 27 = 0$

Step 4.2

Solve $y^{2} - 27 = 0$ for $y$ .

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Step 4.2.1

Add $27$ to both sides of the equation.

$y^{2} = 27$

Step 4.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{27}$

Step 4.2.3

Simplify $\pm \sqrt{27}$ .

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Step 4.2.3.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.3.1.1

Factor $9$ out of $27$ .

$y = \pm \sqrt{9 (3)}$

Step 4.2.3.1.2

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{3^{2} \cdot 3}$

Step 4.2.3.2

Pull terms out from under the radical.

$y = \pm 3 \sqrt{3}$

Step 4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 3 \sqrt{3}$

Step 4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 3 \sqrt{3}$

Step 4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 3 \sqrt{3}, - 3 \sqrt{3}$

Step 5

The final solution is all the values that make $y (y^{2} - 27) = 0$ true.

$y = 0, 3 \sqrt{3}, - 3 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = 0, 3 \sqrt{3}, - 3 \sqrt{3}$

Decimal Form:

$y = 0, 5.19615242 \dots, - 5.19615242 \dots$