Solve by Factoring 125x^3+343=0

$125 x^{3} + 343 = 0$

Step 1

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

${(5 x)}^{3} + 343 = 0$

Step 2

Rewrite $343$ as $7^{3}$ .

${(5 x)}^{3} + 7^{3} = 0$

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 $b = 7$ .

$(5 x + 7) ({(5 x)}^{2} - 5 x \cdot 7 + 7^{2}) = 0$

Step 4

Simplify.

Apply the product rule to $5 x$ .

$(5 x + 7) (5^{2} x^{2} - 5 x \cdot 7 + 7^{2}) = 0$

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

$(5 x + 7) (25 x^{2} - 5 x \cdot 7 + 7^{2}) = 0$

Multiply $5$ by $- 1$ .

$(5 x + 7) (25 x^{2} - 5 x \cdot 7 + 7^{2}) = 0$

Multiply $7$ by $- 5$ .

$(5 x + 7) (25 x^{2} - 35 x + 7^{2}) = 0$

Raise $7$ to the power of $2$ .

$(5 x + 7) (25 x^{2} - 35 x + 49) = 0$

Step 5

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

$5 x + 7 = 0$

$25 x^{2} - 35 x + 49 = 0$

Step 6

Set $5 x + 7$ equal to $0$ and solve for $x$ .

Set $5 x + 7$ equal to $0$ .

$5 x + 7 = 0$

Solve $5 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$5 x = - 7$

Divide each term in $5 x = - 7$ by $5$ and simplify.

Divide each term in $5 x = - 7$ by $5$ .

$\frac{5 x}{5} = \frac{- 7}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 7}{5}$

Divide $x$ by $1$ .

$x = \frac{- 7}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{5}$

Step 7

Set $25 x^{2} - 35 x + 49$ equal to $0$ and solve for $x$ .

Set $25 x^{2} - 35 x + 49$ equal to $0$ .

$25 x^{2} - 35 x + 49 = 0$

Solve $25 x^{2} - 35 x + 49 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 25$ , $b = - 35$ , and $c = 49$ into the quadratic formula and solve for $x$ .

$\frac{35 \pm \sqrt{{(- 35)}^{2} - 4 \cdot (25 \cdot 49)}}{2 \cdot 25}$

Simplify.

Simplify the numerator.

Raise $- 35$ to the power of $2$ .

$x = \frac{35 \pm \sqrt{1225 - 4 \cdot 25 \cdot 49}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 49$ .

Multiply $- 4$ by $25$ .

$x = \frac{35 \pm \sqrt{1225 - 100 \cdot 49}}{2 \cdot 25}$

Multiply $- 100$ by $49$ .

$x = \frac{35 \pm \sqrt{1225 - 4900}}{2 \cdot 25}$

Subtract $4900$ from $1225$ .

$x = \frac{35 \pm \sqrt{- 3675}}{2 \cdot 25}$

Rewrite $- 3675$ as $- 1 (3675)$ .

$x = \frac{35 \pm \sqrt{- 1 \cdot 3675}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (3675)}$ as $\sqrt{- 1} \cdot \sqrt{3675}$ .

$x = \frac{35 \pm \sqrt{- 1} \cdot \sqrt{3675}}{2 \cdot 25}$

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

$x = \frac{35 \pm i \cdot \sqrt{3675}}{2 \cdot 25}$

Rewrite $3675$ as $35^{2} \cdot 3$ .

Factor $1225$ out of $3675$ .

$x = \frac{35 \pm i \cdot \sqrt{1225 (3)}}{2 \cdot 25}$

Rewrite $1225$ as $35^{2}$ .

$x = \frac{35 \pm i \cdot \sqrt{35^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{35 \pm i \cdot (35 \sqrt{3})}{2 \cdot 25}$

Move $35$ to the left of $i$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{50}$

Simplify $\frac{35 \pm 35 i \sqrt{3}}{50}$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{10}$

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

Simplify the numerator.

Raise $- 35$ to the power of $2$ .

$x = \frac{35 \pm \sqrt{1225 - 4 \cdot 25 \cdot 49}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 49$ .

Multiply $- 4$ by $25$ .

$x = \frac{35 \pm \sqrt{1225 - 100 \cdot 49}}{2 \cdot 25}$

Multiply $- 100$ by $49$ .

$x = \frac{35 \pm \sqrt{1225 - 4900}}{2 \cdot 25}$

Subtract $4900$ from $1225$ .

$x = \frac{35 \pm \sqrt{- 3675}}{2 \cdot 25}$

Rewrite $- 3675$ as $- 1 (3675)$ .

$x = \frac{35 \pm \sqrt{- 1 \cdot 3675}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (3675)}$ as $\sqrt{- 1} \cdot \sqrt{3675}$ .

$x = \frac{35 \pm \sqrt{- 1} \cdot \sqrt{3675}}{2 \cdot 25}$

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

$x = \frac{35 \pm i \cdot \sqrt{3675}}{2 \cdot 25}$

Rewrite $3675$ as $35^{2} \cdot 3$ .

Factor $1225$ out of $3675$ .

$x = \frac{35 \pm i \cdot \sqrt{1225 (3)}}{2 \cdot 25}$

Rewrite $1225$ as $35^{2}$ .

$x = \frac{35 \pm i \cdot \sqrt{35^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{35 \pm i \cdot (35 \sqrt{3})}{2 \cdot 25}$

Move $35$ to the left of $i$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{50}$

Simplify $\frac{35 \pm 35 i \sqrt{3}}{50}$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{10}$

Change the $\pm$ to $+$ .

$x = \frac{7 + 7 i \sqrt{3}}{10}$

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

Simplify the numerator.

Raise $- 35$ to the power of $2$ .

$x = \frac{35 \pm \sqrt{1225 - 4 \cdot 25 \cdot 49}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 49$ .

Multiply $- 4$ by $25$ .

$x = \frac{35 \pm \sqrt{1225 - 100 \cdot 49}}{2 \cdot 25}$

Multiply $- 100$ by $49$ .

$x = \frac{35 \pm \sqrt{1225 - 4900}}{2 \cdot 25}$

Subtract $4900$ from $1225$ .

$x = \frac{35 \pm \sqrt{- 3675}}{2 \cdot 25}$

Rewrite $- 3675$ as $- 1 (3675)$ .

$x = \frac{35 \pm \sqrt{- 1 \cdot 3675}}{2 \cdot 25}$

Rewrite $\sqrt{- 1 (3675)}$ as $\sqrt{- 1} \cdot \sqrt{3675}$ .

$x = \frac{35 \pm \sqrt{- 1} \cdot \sqrt{3675}}{2 \cdot 25}$

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

$x = \frac{35 \pm i \cdot \sqrt{3675}}{2 \cdot 25}$

Rewrite $3675$ as $35^{2} \cdot 3$ .

Factor $1225$ out of $3675$ .

$x = \frac{35 \pm i \cdot \sqrt{1225 (3)}}{2 \cdot 25}$

Rewrite $1225$ as $35^{2}$ .

$x = \frac{35 \pm i \cdot \sqrt{35^{2} \cdot 3}}{2 \cdot 25}$

Pull terms out from under the radical.

$x = \frac{35 \pm i \cdot (35 \sqrt{3})}{2 \cdot 25}$

Move $35$ to the left of $i$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$x = \frac{35 \pm 35 i \sqrt{3}}{50}$

Simplify $\frac{35 \pm 35 i \sqrt{3}}{50}$ .

$x = \frac{7 \pm 7 i \sqrt{3}}{10}$

Change the $\pm$ to $-$ .

$x = \frac{7 - 7 i \sqrt{3}}{10}$

The final answer is the combination of both solutions.

$x = \frac{7 + 7 i \sqrt{3}}{10}, \frac{7 - 7 i \sqrt{3}}{10}$

Step 8

The final solution is all the values that make $(5 x + 7) (25 x^{2} - 35 x + 49) = 0$ true.

$x = - \frac{7}{5}, \frac{7 + 7 i \sqrt{3}}{10}, \frac{7 - 7 i \sqrt{3}}{10}$

Solve by Factoring 125x^3+343=0

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