Solve by Factoring 125x^3+216=0

$125 x^{3} + 216 = 0$

Step 1

Rewrite $125 x^{3}$ as ${(5 x)}^{3}$ .

${(5 x)}^{3} + 216 = 0$

Step 2

Rewrite $216$ as $6^{3}$ .

${(5 x)}^{3} + 6^{3} = 0$

Step 3

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 $a = 5 x$ and $b = 6$ .

$(5 x + 6) ({(5 x)}^{2} - 5 x \cdot 6 + 6^{2}) = 0$

Step 4

Simplify.

Apply the product rule to $5 x$ .

$(5 x + 6) (5^{2} x^{2} - 5 x \cdot 6 + 6^{2}) = 0$

Raise $5$ to the power of $2$ .

$(5 x + 6) (25 x^{2} - 5 x \cdot 6 + 6^{2}) = 0$

Multiply $5$ by $- 1$ .

$(5 x + 6) (25 x^{2} - 5 x \cdot 6 + 6^{2}) = 0$

Multiply $6$ by $- 5$ .

$(5 x + 6) (25 x^{2} - 30 x + 6^{2}) = 0$

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

$(5 x + 6) (25 x^{2} - 30 x + 36) = 0$

Step 5

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

$5 x + 6 = 0$

$25 x^{2} - 30 x + 36 = 0$

Step 6

Set $5 x + 6$ equal to $0$ and solve for $x$ .

Set $5 x + 6$ equal to $0$ .

$5 x + 6 = 0$

Solve $5 x + 6 = 0$ for $x$ .

Subtract $6$ from both sides of the equation.

$5 x = - 6$

Divide each term in $5 x = - 6$ by $5$ and simplify.

Divide each term in $5 x = - 6$ by $5$ .

$\frac{5 x}{5} = \frac{- 6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 6}{5}$

Divide $x$ by $1$ .

$x = \frac{- 6}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{6}{5}$

Step 7

Set $25 x^{2} - 30 x + 36$ equal to $0$ and solve for $x$ .

Set $25 x^{2} - 30 x + 36$ equal to $0$ .

$25 x^{2} - 30 x + 36 = 0$

Solve $25 x^{2} - 30 x + 36 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 25$ , $b = - 30$ , and $c = 36$ into the quadratic formula and solve for $x$ .

$\frac{30 \pm \sqrt{{(- 30)}^{2} - 4 \cdot (25 \cdot 36)}}{2 \cdot 25}$

Simplify.

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 25 \cdot 36}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 36$ .

Multiply $- 4$ by $25$ .

$x = \frac{30 \pm \sqrt{900 - 100 \cdot 36}}{2 \cdot 25}$

Multiply $- 100$ by $36$ .

$x = \frac{30 \pm \sqrt{900 - 3600}}{2 \cdot 25}$

Subtract $3600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 2700}}{2 \cdot 25}$

Rewrite $- 2700$ as $- 1 (2700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 2700}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (2700)}$ as $\sqrt{- 1} \cdot \sqrt{2700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{2700}}{2 \cdot 25}$

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

$x = \frac{30 \pm i \cdot \sqrt{2700}}{2 \cdot 25}$

Rewrite $2700$ as $30^{2} \cdot 3$ .

Factor $900$ out of $2700$ .

$x = \frac{30 \pm i \cdot \sqrt{900 (3)}}{2 \cdot 25}$

Rewrite $900$ as $30^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{30^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (30 \sqrt{3})}{2 \cdot 25}$

Move $30$ to the left of $i$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{50}$

Simplify $\frac{30 \pm 30 i \sqrt{3}}{50}$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{5}$

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

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 25 \cdot 36}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 36$ .

Multiply $- 4$ by $25$ .

$x = \frac{30 \pm \sqrt{900 - 100 \cdot 36}}{2 \cdot 25}$

Multiply $- 100$ by $36$ .

$x = \frac{30 \pm \sqrt{900 - 3600}}{2 \cdot 25}$

Subtract $3600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 2700}}{2 \cdot 25}$

Rewrite $- 2700$ as $- 1 (2700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 2700}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (2700)}$ as $\sqrt{- 1} \cdot \sqrt{2700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{2700}}{2 \cdot 25}$

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

$x = \frac{30 \pm i \cdot \sqrt{2700}}{2 \cdot 25}$

Rewrite $2700$ as $30^{2} \cdot 3$ .

Factor $900$ out of $2700$ .

$x = \frac{30 \pm i \cdot \sqrt{900 (3)}}{2 \cdot 25}$

Rewrite $900$ as $30^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{30^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (30 \sqrt{3})}{2 \cdot 25}$

Move $30$ to the left of $i$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{50}$

Simplify $\frac{30 \pm 30 i \sqrt{3}}{50}$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{5}$

Change the $\pm$ to $+$ .

$x = \frac{3 + 3 i \sqrt{3}}{5}$

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

Simplify the numerator.

Raise $- 30$ to the power of $2$ .

$x = \frac{30 \pm \sqrt{900 - 4 \cdot 25 \cdot 36}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 36$ .

Multiply $- 4$ by $25$ .

$x = \frac{30 \pm \sqrt{900 - 100 \cdot 36}}{2 \cdot 25}$

Multiply $- 100$ by $36$ .

$x = \frac{30 \pm \sqrt{900 - 3600}}{2 \cdot 25}$

Subtract $3600$ from $900$ .

$x = \frac{30 \pm \sqrt{- 2700}}{2 \cdot 25}$

Rewrite $- 2700$ as $- 1 (2700)$ .

$x = \frac{30 \pm \sqrt{- 1 \cdot 2700}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (2700)}$ as $\sqrt{- 1} \cdot \sqrt{2700}$ .

$x = \frac{30 \pm \sqrt{- 1} \cdot \sqrt{2700}}{2 \cdot 25}$

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

$x = \frac{30 \pm i \cdot \sqrt{2700}}{2 \cdot 25}$

Rewrite $2700$ as $30^{2} \cdot 3$ .

Factor $900$ out of $2700$ .

$x = \frac{30 \pm i \cdot \sqrt{900 (3)}}{2 \cdot 25}$

Rewrite $900$ as $30^{2}$ .

$x = \frac{30 \pm i \cdot \sqrt{30^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{30 \pm i \cdot (30 \sqrt{3})}{2 \cdot 25}$

Move $30$ to the left of $i$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{30 \pm 30 i \sqrt{3}}{50}$

Simplify $\frac{30 \pm 30 i \sqrt{3}}{50}$ .

$x = \frac{3 \pm 3 i \sqrt{3}}{5}$

Change the $\pm$ to $-$ .

$x = \frac{3 - 3 i \sqrt{3}}{5}$

The final answer is the combination of both solutions.

$x = \frac{3 + 3 i \sqrt{3}}{5}, \frac{3 - 3 i \sqrt{3}}{5}$

Step 8

The final solution is all the values that make $(5 x + 6) (25 x^{2} - 30 x + 36) = 0$ true.

$x = - \frac{6}{5}, \frac{3 + 3 i \sqrt{3}}{5}, \frac{3 - 3 i \sqrt{3}}{5}$

Solve by Factoring 125x^3+216=0

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