Solve for x -2x^3+14x^2-8x+56=0

$- 2 x^{3} + 14 x^{2} - 8 x + 56 = 0$

Step 1

Factor the left side of the equation.

Factor $2$ out of $- 2 x^{3} + 14 x^{2} - 8 x + 56$ .

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

$2 (- x^{3}) + 14 x^{2} - 8 x + 56 = 0$

Factor $2$ out of $14 x^{2}$ .

$2 (- x^{3}) + 2 (7 x^{2}) - 8 x + 56 = 0$

Factor $2$ out of $- 8 x$ .

$2 (- x^{3}) + 2 (7 x^{2}) + 2 (- 4 x) + 56 = 0$

Factor $2$ out of $56$ .

$2 (- x^{3}) + 2 (7 x^{2}) + 2 (- 4 x) + 2 (28) = 0$

Factor $2$ out of $2 (- x^{3}) + 2 (7 x^{2})$ .

$2 (- x^{3} + 7 x^{2}) + 2 (- 4 x) + 2 (28) = 0$

Factor $2$ out of $2 (- x^{3} + 7 x^{2}) + 2 (- 4 x)$ .

$2 (- x^{3} + 7 x^{2} - 4 x) + 2 (28) = 0$

Factor $2$ out of $2 (- x^{3} + 7 x^{2} - 4 x) + 2 (28)$ .

$2 (- x^{3} + 7 x^{2} - 4 x + 28) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((- x^{3} + 7 x^{2}) - 4 x + 28) = 0$

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

$2 (x^{2} (- x + 7) + 4 (- x + 7)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $- x + 7$ .

$2 ((- x + 7) (x^{2} + 4)) = 0$

Remove unnecessary parentheses.

$2 (- x + 7) (x^{2} + 4) = 0$

Step 2

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

$- x + 7 = 0$

$x^{2} + 4 = 0$

Step 3

Set $- x + 7$ equal to $0$ and solve for $x$ .

Set $- x + 7$ equal to $0$ .

$- x + 7 = 0$

Solve $- x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$- x = - 7$

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

Step 4

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

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

$x^{2} + 4 = 0$

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

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

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

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

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

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

$x = \pm i \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{2^{2}}$

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

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

Step 5

The final solution is all the values that make $2 (- x + 7) (x^{2} + 4) = 0$ true.

$x = 7, 2 i, - 2 i$

Solve for x -2x^3+14x^2-8x+56=0

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