Find All Complex Solutions x^3+2x^2+36x=-72

$x^{3} + 2 x^{2} + 36 x = - 72$

Step 1

Add $72$ to both sides of the equation.

$x^{3} + 2 x^{2} + 36 x + 72 = 0$

Step 2

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x^{3} + 2 x^{2}) + 36 x + 72 = 0$

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

$x^{2} (x + 2) + 36 (x + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$(x + 2) (x^{2} + 36) = 0$

Step 3

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

$x + 2 = 0$

$x^{2} + 36 = 0$

Step 4

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 5

Set $x^{2} + 36$ equal to $0$ and solve for $x$ .

Set $x^{2} + 36$ equal to $0$ .

$x^{2} + 36 = 0$

Solve $x^{2} + 36 = 0$ for $x$ .

Subtract $36$ from both sides of the equation.

$x^{2} = - 36$

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

$x = \pm \sqrt{- 36}$

Simplify $\pm \sqrt{- 36}$ .

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

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

$x = \pm i \cdot \sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

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

$x = \pm i \cdot 6$

Move $6$ to the left of $i$ .

$x = \pm 6 i$

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.

$x = 6 i$

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

$x = - 6 i$

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

$x = 6 i, - 6 i$

Step 6

The final solution is all the values that make $(x + 2) (x^{2} + 36) = 0$ true.

$x = - 2, 6 i, - 6 i$

Find All Complex Solutions x^3+2x^2+36x=-72

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