Solve by Factoring 3x^3+12x^2-9x=3x

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

Step 1

Subtract $3 x$ from both sides of the equation.

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

Step 2

Subtract $3 x$ from $- 9 x$ .

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

Step 3

Factor $3 x$ out of $3 x^{3} + 12 x^{2} - 12 x$ .

Factor $3 x$ out of $3 x^{3}$ .

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

Factor $3 x$ out of $12 x^{2}$ .

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

Factor $3 x$ out of $- 12 x$ .

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

Factor $3 x$ out of $3 x (x^{2}) + 3 x (4 x)$ .

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

Factor $3 x$ out of $3 x (x^{2} + 4 x) + 3 x (- 4)$ .

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

Step 4

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

$x = 0$

$x^{2} + 4 x - 4 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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

$x^{2} + 4 x - 4 = 0$

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 4$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

$x = - 2 \pm 2 \sqrt{2}$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

$x = - 2 \pm 2 \sqrt{2}$

Change the $\pm$ to $+$ .

$x = - 2 + 2 \sqrt{2}$

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

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4$ by $- 4$ .

$x = \frac{- 4 \pm \sqrt{16 + 16}}{2 \cdot 1}$

Add $16$ and $16$ .

$x = \frac{- 4 \pm \sqrt{32}}{2 \cdot 1}$

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x = \frac{- 4 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 4 \sqrt{2}}{2}$

Simplify $\frac{- 4 \pm 4 \sqrt{2}}{2}$ .

$x = - 2 \pm 2 \sqrt{2}$

Change the $\pm$ to $-$ .

$x = - 2 - 2 \sqrt{2}$

The final answer is the combination of both solutions.

$x = - 2 + 2 \sqrt{2}, - 2 - 2 \sqrt{2}$

Step 7

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

$x = 0, - 2 + 2 \sqrt{2}, - 2 - 2 \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, - 2 + 2 \sqrt{2}, - 2 - 2 \sqrt{2}$

Decimal Form:

$x = 0, 0.82842712 \dots, - 4.82842712 \dots$

Solve by Factoring 3x^3+12x^2-9x=3x

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