Solve for y y^3+4y-3y^2-12=0

$y^{3} + 4 y - 3 y^{2} - 12 = 0$

Step 1

Factor the left side of the equation.

Reorder terms.

$y^{3} - 3 y^{2} + 4 y - 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(y^{3} - 3 y^{2}) + 4 y - 12 = 0$

Factor out the greatest common factor (GCF) from each group.

$y^{2} (y - 3) + 4 (y - 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $y - 3$ .

$(y - 3) (y^{2} + 4) = 0$

Step 2

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

$y - 3 = 0$

$y^{2} + 4 = 0$

Step 3

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

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

$y - 3 = 0$

Add $3$ to both sides of the equation.

$y = 3$

Step 4

Set $y^{2} + 4$ equal to $0$ and solve for $y$ .

Set $y^{2} + 4$ equal to $0$ .

$y^{2} + 4 = 0$

Solve $y^{2} + 4 = 0$ for $y$ .

Subtract $4$ from both sides of the equation.

$y^{2} = - 4$

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

$y = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$y = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$y = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

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

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

$y = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$y = \pm 2 i$

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

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

$y = 2 i$

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

$y = - 2 i$

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

$y = 2 i, - 2 i$

Step 5

The final solution is all the values that make $(y - 3) (y^{2} + 4) = 0$ true.

$y = 3, 2 i, - 2 i$

Solve for y y^3+4y-3y^2-12=0

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