Find All Complex Solutions 5y^3-3y^2+8y=0

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

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

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

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

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

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

Factor $y$ out of $8 y$ .

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

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

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

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

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

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

$y = 0$

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

Set $y$ equal to $0$ .

$y = 0$

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

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

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

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = - 3$ , and $c = 8$ into the quadratic formula and solve for $y$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $5$ .

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

Multiply $- 20$ by $8$ .

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

Subtract $160$ from $9$ .

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

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

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4$ by $5$ .

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

Multiply $- 20$ by $8$ .

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

Subtract $160$ from $9$ .

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

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

Change the $\pm$ to $+$ .

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

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

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

Multiply $- 4$ by $5$ .

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

Multiply $- 20$ by $8$ .

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

Subtract $160$ from $9$ .

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

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

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

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

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

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

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

Multiply $2$ by $5$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

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

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

Find All Complex Solutions 5y^3-3y^2+8y=0

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