Solve by Factoring y^3+9y-y^2-9=0

$y^{3} + 9 y - y^{2} - 9 = 0$

Step 1

Reorder terms.

$y^{3} - y^{2} + 9 y - 9 = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(y^{3} - y^{2}) + 9 y - 9 = 0$

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

$y^{2} (y - 1) + 9 (y - 1) = 0$

Step 3

Factor the polynomial by factoring out the greatest common factor, $y - 1$ .

$(y - 1) (y^{2} + 9) = 0$

Step 4

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

$y - 1 = 0$

$y^{2} + 9 = 0$

Step 5

Set $y - 1$ equal to $0$ and solve for $y$ .

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

Add $1$ to both sides of the equation.

$y = 1$

Step 6

Set $y^{2} + 9$ equal to $0$ and solve for $y$ .

Set $y^{2} + 9$ equal to $0$ .

$y^{2} + 9 = 0$

Solve $y^{2} + 9 = 0$ for $y$ .

Subtract $9$ from both sides of the equation.

$y^{2} = - 9$

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

$y = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$y = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$y = \pm i \cdot \sqrt{9}$

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

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

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

$y = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$y = \pm 3 i$

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.

$y = 3 i$

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

$y = - 3 i$

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

$y = 3 i, - 3 i$

Step 7

The final solution is all the values that make $(y - 1) (y^{2} + 9) = 0$ true.

$y = 1, 3 i, - 3 i$

Solve by Factoring y^3+9y-y^2-9=0

Related Questions

Support LCMGCF.com by disabling your adblocker.