Solve by Factoring 9y^3+8=4y+18y^2

$9 y^{3} + 8 = 4 y + 18 y^{2}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $4 y$ from both sides of the equation.

$9 y^{3} + 8 - 4 y = 18 y^{2}$

Subtract $18 y^{2}$ from both sides of the equation.

$9 y^{3} + 8 - 4 y - 18 y^{2} = 0$

Step 2

Reorder terms.

$9 y^{3} - 18 y^{2} - 4 y + 8 = 0$

Step 3

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 y^{3} - 18 y^{2}) - 4 y + 8 = 0$

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

$9 y^{2} (y - 2) - 4 (y - 2) = 0$

Step 4

Factor the polynomial by factoring out the greatest common factor, $y - 2$ .

$(y - 2) (9 y^{2} - 4) = 0$

Step 5

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

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

Step 6

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

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

Step 7

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 y$ and $b = 2$ .

$(y - 2) ((3 y + 2) (3 y - 2)) = 0$

Remove unnecessary parentheses.

$(y - 2) (3 y + 2) (3 y - 2) = 0$

Step 8

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

$y - 2 = 0$

$3 y + 2 = 0$

$3 y - 2 = 0$

Step 9

Set $y - 2$ equal to $0$ and solve for $y$ .

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Add $2$ to both sides of the equation.

$y = 2$

Step 10

Set $3 y + 2$ equal to $0$ and solve for $y$ .

Set $3 y + 2$ equal to $0$ .

$3 y + 2 = 0$

Solve $3 y + 2 = 0$ for $y$ .

Subtract $2$ from both sides of the equation.

$3 y = - 2$

Divide each term in $3 y = - 2$ by $3$ and simplify.

Divide each term in $3 y = - 2$ by $3$ .

$\frac{3 y}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 2}{3}$

Divide $y$ by $1$ .

$y = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{2}{3}$

Step 11

Set $3 y - 2$ equal to $0$ and solve for $y$ .

Set $3 y - 2$ equal to $0$ .

$3 y - 2 = 0$

Solve $3 y - 2 = 0$ for $y$ .

Add $2$ to both sides of the equation.

$3 y = 2$

Divide each term in $3 y = 2$ by $3$ and simplify.

Divide each term in $3 y = 2$ by $3$ .

$\frac{3 y}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{2}{3}$

Divide $y$ by $1$ .

$y = \frac{2}{3}$

Step 12

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

$y = 2, - \frac{2}{3}, \frac{2}{3}$

Solve by Factoring 9y^3+8=4y+18y^2

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