Algebra Examples

$5 y^{3} - 3 y^{2} + 8 y = 0$

Step 1

Factor $y$ out of $5 y^{3} - 3 y^{2} + 8 y$ .

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Step 1.1

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

$y (5 y^{2}) - 3 y^{2} + 8 y = 0$

Step 1.2

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

$y (5 y^{2}) + y (- 3 y) + 8 y = 0$

Step 1.3

Factor $y$ out of $8 y$ .

$y (5 y^{2}) + y (- 3 y) + y \cdot 8 = 0$

Step 1.4

Factor $y$ out of $y (5 y^{2}) + y (- 3 y)$ .

$y (5 y^{2} - 3 y) + y \cdot 8 = 0$

Step 1.5

Factor $y$ out of $y (5 y^{2} - 3 y) + y \cdot 8$ .

$y (5 y^{2} - 3 y + 8) = 0$

Step 2

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

$y = 0$

$5 y^{2} - 3 y + 8 = 0$

Step 3

Set $y$ equal to $0$ .

$y = 0$

Step 4

Set $5 y^{2} - 3 y + 8$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $5 y^{2} - 3 y + 8$ equal to $0$ .

$5 y^{2} - 3 y + 8 = 0$

Step 4.2

Solve $5 y^{2} - 3 y + 8 = 0$ 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 $a = 5$ , $b = - 3$ , and $c = 8$ into the quadratic formula and solve for $y$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (5 \cdot 8)}}{2 \cdot 5}$

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 $- 3$ to the power of $2$ .

$y = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 5 \cdot 8$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{3 \pm \sqrt{9 - 20 \cdot 8}}{2 \cdot 5}$

Step 4.2.3.1.2.2

Multiply $- 20$ by $8$ .

$y = \frac{3 \pm \sqrt{9 - 160}}{2 \cdot 5}$

Step 4.2.3.1.3

Subtract $160$ from $9$ .

$y = \frac{3 \pm \sqrt{- 151}}{2 \cdot 5}$

Step 4.2.3.1.4

Rewrite $- 151$ as $- 1 (151)$ .

$y = \frac{3 \pm \sqrt{- 1 \cdot 151}}{2 \cdot 5}$

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (151)}$ as $\sqrt{- 1} \cdot \sqrt{151}$ .

$y = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{151}}{2 \cdot 5}$

Step 4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{3 \pm i \sqrt{151}}{2 \cdot 5}$

Step 4.2.3.2

Multiply $2$ by $5$ .

$y = \frac{3 \pm i \sqrt{151}}{10}$

Step 4.2.4

The final answer is the combination of both solutions.

$y = \frac{3 + i \sqrt{151}}{10}, \frac{3 - i \sqrt{151}}{10}$

Step 5

The final solution is all the values that make $y (5 y^{2} - 3 y + 8) = 0$ true.

$y = 0, \frac{3 + i \sqrt{151}}{10}, \frac{3 - i \sqrt{151}}{10}$