Solve by Factoring 108x^3+37=5

$108 x^{3} + 37 = 5$

Step 1

Subtract $5$ from both sides of the equation.

$108 x^{3} + 37 - 5 = 0$

Step 2

Subtract $5$ from $37$ .

$108 x^{3} + 32 = 0$

Step 3

Factor $4$ out of $108 x^{3} + 32$ .

Factor $4$ out of $108 x^{3}$ .

$4 (27 x^{3}) + 32 = 0$

Factor $4$ out of $32$ .

$4 (27 x^{3}) + 4 (8) = 0$

Factor $4$ out of $4 (27 x^{3}) + 4 (8)$ .

$4 (27 x^{3} + 8) = 0$

Step 4

Rewrite $27 x^{3}$ as ${(3 x)}^{3}$ .

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

Step 5

Rewrite $8$ as $2^{3}$ .

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

Step 6

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

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

Step 7

Factor.

Simplify.

Apply the product rule to $3 x$ .

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

Raise $3$ to the power of $2$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $2$ by $- 3$ .

$4 ((3 x + 2) (9 x^{2} - 6 x + 2^{2})) = 0$

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

$4 ((3 x + 2) (9 x^{2} - 6 x + 4)) = 0$

Remove unnecessary parentheses.

$4 (3 x + 2) (9 x^{2} - 6 x + 4) = 0$

Step 8

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

$3 x + 2 = 0$

$9 x^{2} - 6 x + 4 = 0$

Step 9

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

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

$3 x + 2 = 0$

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

Subtract $2$ from both sides of the equation.

$3 x = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 10

Set $9 x^{2} - 6 x + 4$ equal to $0$ and solve for $x$ .

Set $9 x^{2} - 6 x + 4$ equal to $0$ .

$9 x^{2} - 6 x + 4 = 0$

Solve $9 x^{2} - 6 x + 4 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 9$ , $b = - 6$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (9 \cdot 4)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot 4}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 4$ .

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot 4}}{2 \cdot 9}$

Multiply $- 36$ by $4$ .

$x = \frac{6 \pm \sqrt{36 - 144}}{2 \cdot 9}$

Subtract $144$ from $36$ .

$x = \frac{6 \pm \sqrt{- 108}}{2 \cdot 9}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 108}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{108}}{2 \cdot 9}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{6 \pm i \cdot \sqrt{36 (3)}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{6 \pm i \cdot (6 \sqrt{3})}{2 \cdot 9}$

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

$x = \frac{6 \pm 6 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 i \sqrt{3}}{18}$

Simplify $\frac{6 \pm 6 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{3}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot 4}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 4$ .

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot 4}}{2 \cdot 9}$

Multiply $- 36$ by $4$ .

$x = \frac{6 \pm \sqrt{36 - 144}}{2 \cdot 9}$

Subtract $144$ from $36$ .

$x = \frac{6 \pm \sqrt{- 108}}{2 \cdot 9}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 108}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{108}}{2 \cdot 9}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{6 \pm i \cdot \sqrt{36 (3)}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{6 \pm i \cdot (6 \sqrt{3})}{2 \cdot 9}$

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

$x = \frac{6 \pm 6 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 i \sqrt{3}}{18}$

Simplify $\frac{6 \pm 6 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{3}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{3}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot 4}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 4$ .

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot 4}}{2 \cdot 9}$

Multiply $- 36$ by $4$ .

$x = \frac{6 \pm \sqrt{36 - 144}}{2 \cdot 9}$

Subtract $144$ from $36$ .

$x = \frac{6 \pm \sqrt{- 108}}{2 \cdot 9}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 108}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{108}}{2 \cdot 9}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{6 \pm i \cdot \sqrt{36 (3)}}{2 \cdot 9}$

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

$x = \frac{6 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot 9}$

Pull terms out from under the radical.

$x = \frac{6 \pm i \cdot (6 \sqrt{3})}{2 \cdot 9}$

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

$x = \frac{6 \pm 6 i \sqrt{3}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 i \sqrt{3}}{18}$

Simplify $\frac{6 \pm 6 i \sqrt{3}}{18}$ .

$x = \frac{1 \pm i \sqrt{3}}{3}$

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{3}$

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{3}, \frac{1 - i \sqrt{3}}{3}$

Step 11

The final solution is all the values that make $4 (3 x + 2) (9 x^{2} - 6 x + 4) = 0$ true.

$x = - \frac{2}{3}, \frac{1 + i \sqrt{3}}{3}, \frac{1 - i \sqrt{3}}{3}$

Solve by Factoring 108x^3+37=5

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