Algebra Examples

$- 600 = \frac{2}{5} k^{3} + 750$

Step 1

Rewrite the equation as $\frac{2}{5} k^{3} + 750 = - 600$ .

$\frac{2}{5} k^{3} + 750 = - 600$

Step 2

Combine $\frac{2}{5}$ and $k^{3}$ .

$\frac{2 k^{3}}{5} + 750 = - 600$

Step 3

Move all terms not containing $k$ to the right side of the equation.

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Step 3.1

Subtract $750$ from both sides of the equation.

$\frac{2 k^{3}}{5} = - 600 - 750$

Step 3.2

Subtract $750$ from $- 600$ .

$\frac{2 k^{3}}{5} = - 1350$

Step 4

Multiply both sides of the equation by $\frac{5}{2}$ .

$\frac{5}{2} \cdot \frac{2 k^{3}}{5} = \frac{5}{2} \cdot - 1350$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{5}{2} \cdot \frac{2 k^{3}}{5}$ .

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Step 5.1.1.1

Combine.

$\frac{5 (2 k^{3})}{2 \cdot 5} = \frac{5}{2} \cdot - 1350$

Step 5.1.1.2

Cancel the common factor of $5$ .

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Step 5.1.1.2.1

Cancel the common factor.

$\frac{5 (2 k^{3})}{2 \cdot 5} = \frac{5}{2} \cdot - 1350$

Step 5.1.1.2.2

Rewrite the expression.

$\frac{2 k^{3}}{2} = \frac{5}{2} \cdot - 1350$

Step 5.1.1.3

Cancel the common factor of $2$ .

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Step 5.1.1.3.1

Cancel the common factor.

$\frac{2 k^{3}}{2} = \frac{5}{2} \cdot - 1350$

Step 5.1.1.3.2

Divide $k^{3}$ by $1$ .

$k^{3} = \frac{5}{2} \cdot - 1350$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{5}{2} \cdot - 1350$ .

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Step 5.2.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1

Factor $2$ out of $- 1350$ .

$k^{3} = \frac{5}{2} \cdot (2 (- 675))$

Step 5.2.1.1.2

Cancel the common factor.

$k^{3} = \frac{5}{2} \cdot (2 \cdot - 675)$

Step 5.2.1.1.3

Rewrite the expression.

$k^{3} = 5 \cdot - 675$

Step 5.2.1.2

Multiply $5$ by $- 675$ .

$k^{3} = - 3375$

Step 6

Add $3375$ to both sides of the equation.

$k^{3} + 3375 = 0$

Step 7

Factor the left side of the equation.

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Step 7.1

Rewrite $3375$ as $15^{3}$ .

$k^{3} + 15^{3} = 0$

Step 7.2

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 $a = k$ and $b = 15$ .

$(k + 15) (k^{2} - k \cdot 15 + 15^{2}) = 0$

Step 7.3

Simplify.

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Step 7.3.1

Multiply $15$ by $- 1$ .

$(k + 15) (k^{2} - 15 k + 15^{2}) = 0$

Step 7.3.2

Raise $15$ to the power of $2$ .

$(k + 15) (k^{2} - 15 k + 225) = 0$

Step 8

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

$k + 15 = 0$

$k^{2} - 15 k + 225 = 0$

Step 9

Set $k + 15$ equal to $0$ and solve for $k$ .

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Step 9.1

Set $k + 15$ equal to $0$ .

$k + 15 = 0$

Step 9.2

Subtract $15$ from both sides of the equation.

$k = - 15$

Step 10

Set $k^{2} - 15 k + 225$ equal to $0$ and solve for $k$ .

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Step 10.1

Set $k^{2} - 15 k + 225$ equal to $0$ .

$k^{2} - 15 k + 225 = 0$

Step 10.2

Solve $k^{2} - 15 k + 225 = 0$ for $k$ .

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Step 10.2.1

Use the quadratic formula to find the solutions.

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

Step 10.2.2

Substitute the values $a = 1$ , $b = - 15$ , and $c = 225$ into the quadratic formula and solve for $k$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot 225)}}{2 \cdot 1}$

Step 10.2.3

Simplify.

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Step 10.2.3.1

Simplify the numerator.

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Step 10.2.3.1.1

Raise $- 15$ to the power of $2$ .

$k = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 225}}{2 \cdot 1}$

Step 10.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 225$ .

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Step 10.2.3.1.2.1

Multiply $- 4$ by $1$ .

$k = \frac{15 \pm \sqrt{225 - 4 \cdot 225}}{2 \cdot 1}$

Step 10.2.3.1.2.2

Multiply $- 4$ by $225$ .

$k = \frac{15 \pm \sqrt{225 - 900}}{2 \cdot 1}$

Step 10.2.3.1.3

Subtract $900$ from $225$ .

$k = \frac{15 \pm \sqrt{- 675}}{2 \cdot 1}$

Step 10.2.3.1.4

Rewrite $- 675$ as $- 1 (675)$ .

$k = \frac{15 \pm \sqrt{- 1 \cdot 675}}{2 \cdot 1}$

Step 10.2.3.1.5

Rewrite $\sqrt{- 1 (675)}$ as $\sqrt{- 1} \cdot \sqrt{675}$ .

$k = \frac{15 \pm \sqrt{- 1} \cdot \sqrt{675}}{2 \cdot 1}$

Step 10.2.3.1.6

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

$k = \frac{15 \pm i \cdot \sqrt{675}}{2 \cdot 1}$

Step 10.2.3.1.7

Rewrite $675$ as $15^{2} \cdot 3$ .

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Step 10.2.3.1.7.1

Factor $225$ out of $675$ .

$k = \frac{15 \pm i \cdot \sqrt{225 (3)}}{2 \cdot 1}$

Step 10.2.3.1.7.2

Rewrite $225$ as $15^{2}$ .

$k = \frac{15 \pm i \cdot \sqrt{15^{2} \cdot 3}}{2 \cdot 1}$

Step 10.2.3.1.8

Pull terms out from under the radical.

$k = \frac{15 \pm i \cdot (15 \sqrt{3})}{2 \cdot 1}$

Step 10.2.3.1.9

Move $15$ to the left of $i$ .

$k = \frac{15 \pm 15 i \sqrt{3}}{2 \cdot 1}$

Step 10.2.3.2

Multiply $2$ by $1$ .

$k = \frac{15 \pm 15 i \sqrt{3}}{2}$

Step 10.2.4

The final answer is the combination of both solutions.

$k = \frac{15 + 15 i \sqrt{3}}{2}, \frac{15 - 15 i \sqrt{3}}{2}$

Step 11

The final solution is all the values that make $(k + 15) (k^{2} - 15 k + 225) = 0$ true.

$k = - 15, \frac{15 + 15 i \sqrt{3}}{2}, \frac{15 - 15 i \sqrt{3}}{2}$