Basic Math Examples

$0.072 v - \frac{243}{v^{2}} = 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, v^{2}, 1$

Step 1.2

The LCM of one and any expression is the expression.

$v^{2}$

Step 2

Multiply each term in $0.072 v - \frac{243}{v^{2}} = 0$ by $v^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $0.072 v - \frac{243}{v^{2}} = 0$ by $v^{2}$ .

$0.072 v \cdot v^{2} - \frac{243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $v$ by $v^{2}$ by adding the exponents.

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Step 2.2.1.1.1

Move $v^{2}$ .

$0.072 (v^{2} v) - \frac{243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.1.2

Multiply $v^{2}$ by $v$ .

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Step 2.2.1.1.2.1

Raise $v$ to the power of $1$ .

$0.072 (v^{2} v^{1}) - \frac{243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0.072 v^{2 + 1} - \frac{243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.1.3

Add $2$ and $1$ .

$0.072 v^{3} - \frac{243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.2

Cancel the common factor of $v^{2}$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{243}{v^{2}}$ into the numerator.

$0.072 v^{3} + \frac{- 243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.2.2

Cancel the common factor.

$0.072 v^{3} + \frac{- 243}{v^{2}} v^{2} = 0 v^{2}$

Step 2.2.1.2.3

Rewrite the expression.

$0.072 v^{3} - 243 = 0 v^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $v^{2}$ .

$0.072 v^{3} - 243 = 0$

Step 3

Solve the equation.

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Step 3.1

Add $243$ to both sides of the equation.

$0.072 v^{3} = 243$

Step 3.2

Subtract $243$ from both sides of the equation.

$0.072 v^{3} - 243 = 0$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $0.072$ out of $0.072 v^{3} - 243$ .

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Step 3.3.1.1

Factor $0.072$ out of $0.072 v^{3}$ .

$0.072 (v^{3}) - 243 = 0$

Step 3.3.1.2

Factor $0.072$ out of $- 243$ .

$0.072 v^{3} + 0.072 \cdot - 3375 = 0$

Step 3.3.1.3

Factor $0.072$ out of $0.072 v^{3} + 0.072 \cdot - 3375$ .

$0.072 (v^{3} - 3375) = 0$

Step 3.3.2

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

$0.072 (v^{3} - 15^{3}) = 0$

Step 3.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = v$ and $b = 15$ .

$0.072 ((v - 15) (v^{2} + v \cdot 15 + 15^{2})) = 0$

Step 3.3.4

Factor.

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Step 3.3.4.1

Simplify.

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Step 3.3.4.1.1

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

$0.072 ((v - 15) (v^{2} + 15 v + 15^{2})) = 0$

Step 3.3.4.1.2

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

$0.072 ((v - 15) (v^{2} + 15 v + 225)) = 0$

Step 3.3.4.2

Remove unnecessary parentheses.

$0.072 (v - 15) (v^{2} + 15 v + 225) = 0$

Step 3.4

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

$v - 15 = 0$

$v^{2} + 15 v + 225 = 0$

Step 3.5

Set $v - 15$ equal to $0$ and solve for $v$ .

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Step 3.5.1

Set $v - 15$ equal to $0$ .

$v - 15 = 0$

Step 3.5.2

Add $15$ to both sides of the equation.

$v = 15$

Step 3.6

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

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Step 3.6.1

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

$v^{2} + 15 v + 225 = 0$

Step 3.6.2

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

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Step 3.6.2.1

Use the quadratic formula to find the solutions.

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

Step 3.6.2.2

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

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

Step 3.6.2.3

Simplify.

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Step 3.6.2.3.1

Simplify the numerator.

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Step 3.6.2.3.1.1

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

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

Step 3.6.2.3.1.2

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

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Step 3.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 3.6.2.3.1.2.2

Multiply $- 4$ by $225$ .

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

Step 3.6.2.3.1.3

Subtract $900$ from $225$ .

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

Step 3.6.2.3.1.4

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

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

Step 3.6.2.3.1.5

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

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

Step 3.6.2.3.1.6

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

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

Step 3.6.2.3.1.7

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

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Step 3.6.2.3.1.7.1

Factor $225$ out of $675$ .

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

Step 3.6.2.3.1.7.2

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

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

Step 3.6.2.3.1.8

Pull terms out from under the radical.

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

Step 3.6.2.3.1.9

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

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

Step 3.6.2.3.2

Multiply $2$ by $1$ .

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

Step 3.6.2.4

The final answer is the combination of both solutions.

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

Step 3.7

The final solution is all the values that make $0.072 (v - 15) (v^{2} + 15 v + 225) = 0$ true.

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