Precalculus Examples

$y^{5} - 18 y^{4} + 6 y^{3} = 0$

Step 1

Factor $y^{3}$ out of $y^{5} - 18 y^{4} + 6 y^{3}$ .

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Step 1.1

Factor $y^{3}$ out of $y^{5}$ .

$y^{3} y^{2} - 18 y^{4} + 6 y^{3} = 0$

Step 1.2

Factor $y^{3}$ out of $- 18 y^{4}$ .

$y^{3} y^{2} + y^{3} (- 18 y) + 6 y^{3} = 0$

Step 1.3

Factor $y^{3}$ out of $6 y^{3}$ .

$y^{3} y^{2} + y^{3} (- 18 y) + y^{3} \cdot 6 = 0$

Step 1.4

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

$y^{3} (y^{2} - 18 y) + y^{3} \cdot 6 = 0$

Step 1.5

Factor $y^{3}$ out of $y^{3} (y^{2} - 18 y) + y^{3} \cdot 6$ .

$y^{3} (y^{2} - 18 y + 6) = 0$

Step 2

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

$y^{3} = 0$

$y^{2} - 18 y + 6 = 0$

Step 3

Set $y^{3}$ equal to $0$ and solve for $y$ .

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Step 3.1

Set $y^{3}$ equal to $0$ .

$y^{3} = 0$

Step 3.2

Solve $y^{3} = 0$ for $y$ .

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Step 3.2.1

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

$y = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$y = \sqrt[3]{0^{3}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$y = 0$

Step 4

Set $y^{2} - 18 y + 6$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $y^{2} - 18 y + 6$ equal to $0$ .

$y^{2} - 18 y + 6 = 0$

Step 4.2

Solve $y^{2} - 18 y + 6 = 0$ for $y$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2.2

Substitute the values $a = 1$ , $b = - 18$ , and $c = 6$ into the quadratic formula and solve for $y$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise $- 18$ to the power of $2$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $6$ .

$y = \frac{18 \pm \sqrt{324 - 24}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $24$ from $324$ .

$y = \frac{18 \pm \sqrt{300}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $300$ as $10^{2} \cdot 3$ .

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Step 4.2.3.1.4.1

Factor $100$ out of $300$ .

$y = \frac{18 \pm \sqrt{100 (3)}}{2 \cdot 1}$

Step 4.2.3.1.4.2

Rewrite $100$ as $10^{2}$ .

$y = \frac{18 \pm \sqrt{10^{2} \cdot 3}}{2 \cdot 1}$

Step 4.2.3.1.5

Pull terms out from under the radical.

$y = \frac{18 \pm 10 \sqrt{3}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$y = \frac{18 \pm 10 \sqrt{3}}{2}$

Step 4.2.3.3

Simplify $\frac{18 \pm 10 \sqrt{3}}{2}$ .

$y = 9 \pm 5 \sqrt{3}$

Step 4.2.4

The final answer is the combination of both solutions.

$y = 9 + 5 \sqrt{3}, 9 - 5 \sqrt{3}$

Step 5

The final solution is all the values that make $y^{3} (y^{2} - 18 y + 6) = 0$ true.

$y = 0, 9 + 5 \sqrt{3}, 9 - 5 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = 0, 9 + 5 \sqrt{3}, 9 - 5 \sqrt{3}$

Decimal Form:

$y = 0, 17.66025403 \dots, 0.33974596 \dots$