Find All Complex Solutions -2x^4+8x^3+8x^2-32x=0

$- 2 x^{4} + 8 x^{3} + 8 x^{2} - 32 x = 0$

Step 1

Factor the left side of the equation.

Factor $2 x$ out of $- 2 x^{4} + 8 x^{3} + 8 x^{2} - 32 x$ .

Factor $2 x$ out of $- 2 x^{4}$ .

$2 x (- x^{3}) + 8 x^{3} + 8 x^{2} - 32 x = 0$

Factor $2 x$ out of $8 x^{3}$ .

$2 x (- x^{3}) + 2 x (4 x^{2}) + 8 x^{2} - 32 x = 0$

Factor $2 x$ out of $8 x^{2}$ .

$2 x (- x^{3}) + 2 x (4 x^{2}) + 2 x (4 x) - 32 x = 0$

Factor $2 x$ out of $- 32 x$ .

$2 x (- x^{3}) + 2 x (4 x^{2}) + 2 x (4 x) + 2 x (- 16) = 0$

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

$2 x (- x^{3} + 4 x^{2}) + 2 x (4 x) + 2 x (- 16) = 0$

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

$2 x (- x^{3} + 4 x^{2} + 4 x) + 2 x (- 16) = 0$

Factor $2 x$ out of $2 x (- x^{3} + 4 x^{2} + 4 x) + 2 x (- 16)$ .

$2 x (- x^{3} + 4 x^{2} + 4 x - 16) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 x ((- x^{3} + 4 x^{2}) + 4 x - 16) = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $- x + 4$ .

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

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

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

Factor.

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

$2 x ((- x + 4) ((x + 2) (x - 2))) = 0$

Remove unnecessary parentheses.

$2 x ((- x + 4) (x + 2) (x - 2)) = 0$

Remove unnecessary parentheses.

$2 x (- x + 4) (x + 2) (x - 2) = 0$

Step 2

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

$x = 0$

$- x + 4 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $- x + 4$ equal to $0$ and solve for $x$ .

Set $- x + 4$ equal to $0$ .

$- x + 4 = 0$

Solve $- x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

Step 5

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

$x = 0, 4, - 2, 2$

Find All Complex Solutions -2x^4+8x^3+8x^2-32x=0

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