Solve by Factoring 4x^3-12x^2=9x-27

$4 x^{3} - 12 x^{2} = 9 x - 27$

Step 1

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

Subtract $9 x$ from both sides of the equation.

$4 x^{3} - 12 x^{2} - 9 x = - 27$

Add $27$ to both sides of the equation.

$4 x^{3} - 12 x^{2} - 9 x + 27 = 0$

Step 2

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{3} - 12 x^{2}) - 9 x + 27 = 0$

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

$4 x^{2} (x - 3) - 9 (x - 3) = 0$

Step 3

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

$(x - 3) (4 x^{2} - 9) = 0$

Step 4

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

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

Step 5

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

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

Step 6

Factor.

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

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

Remove unnecessary parentheses.

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

Step 7

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

$x - 3 = 0$

$2 x + 3 = 0$

$2 x - 3 = 0$

Step 8

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 9

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

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

$2 x + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$2 x = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 10

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

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

$2 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 11

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

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

Solve by Factoring 4x^3-12x^2=9x-27

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