Algebra Examples

$(3 \cdot 6 \div 2) v + 10 = 3^{2} v + 9$

Step 1

Simplify $3 \cdot 6 \div 2 v + 10$ .

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Step 1.1

Multiply $3$ by $6$ .

$18 \div 2 v + 10 = 3^{2} v + 9$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite the division as a fraction.

$\frac{18}{2 v} + 10 = 3^{2} v + 9$

Step 1.2.2

Cancel the common factor of $18$ and $2$ .

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Step 1.2.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9}{2 v} + 10 = 3^{2} v + 9$

Step 1.2.2.2

Cancel the common factors.

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Step 1.2.2.2.1

Factor $2$ out of $2 v$ .

$\frac{2 \cdot 9}{2 (v)} + 10 = 3^{2} v + 9$

Step 1.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 v} + 10 = 3^{2} v + 9$

Step 1.2.2.2.3

Rewrite the expression.

$\frac{9}{v} + 10 = 3^{2} v + 9$

Step 2

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

$\frac{9}{v} + 10 = 9 v + 9$

Step 3

Subtract $9 v$ from both sides of the equation.

$\frac{9}{v} + 10 - 9 v = 9$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

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

$v, 1, 1, 1$

Step 4.2

The LCM of one and any expression is the expression.

$v$

Step 5

Multiply each term in $\frac{9}{v} + 10 - 9 v = 9$ by $v$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{9}{v} + 10 - 9 v = 9$ by $v$ .

$\frac{9}{v} v + 10 v - 9 v \cdot v = 9 v$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of $v$ .

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Step 5.2.1.1.1

Cancel the common factor.

$\frac{9}{v} v + 10 v - 9 v \cdot v = 9 v$

Step 5.2.1.1.2

Rewrite the expression.

$9 + 10 v - 9 v \cdot v = 9 v$

Step 5.2.1.2

Multiply $v$ by $v$ by adding the exponents.

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Step 5.2.1.2.1

Move $v$ .

$9 + 10 v - 9 (v \cdot v) = 9 v$

Step 5.2.1.2.2

Multiply $v$ by $v$ .

$9 + 10 v - 9 v^{2} = 9 v$

Step 6

Solve the equation.

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Step 6.1

Move all terms containing $v$ to the left side of the equation.

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Step 6.1.1

Subtract $9 v$ from both sides of the equation.

$9 + 10 v - 9 v^{2} - 9 v = 0$

Step 6.1.2

Subtract $9 v$ from $10 v$ .

$9 - 9 v^{2} + v = 0$

Step 6.2

Use the quadratic formula to find the solutions.

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

Step 6.3

Substitute the values $a = - 9$ , $b = 1$ , and $c = 9$ into the quadratic formula and solve for $v$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 9 \cdot 9)}}{2 \cdot - 9}$

Step 6.4

Simplify.

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Step 6.4.1

Simplify the numerator.

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Step 6.4.1.1

One to any power is one.

$v = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 9 \cdot 9}}{2 \cdot - 9}$

Step 6.4.1.2

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

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Step 6.4.1.2.1

Multiply $- 4$ by $- 9$ .

$v = \frac{- 1 \pm \sqrt{1 + 36 \cdot 9}}{2 \cdot - 9}$

Step 6.4.1.2.2

Multiply $36$ by $9$ .

$v = \frac{- 1 \pm \sqrt{1 + 324}}{2 \cdot - 9}$

Step 6.4.1.3

Add $1$ and $324$ .

$v = \frac{- 1 \pm \sqrt{325}}{2 \cdot - 9}$

Step 6.4.1.4

Rewrite $325$ as $5^{2} \cdot 13$ .

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Step 6.4.1.4.1

Factor $25$ out of $325$ .

$v = \frac{- 1 \pm \sqrt{25 (13)}}{2 \cdot - 9}$

Step 6.4.1.4.2

Rewrite $25$ as $5^{2}$ .

$v = \frac{- 1 \pm \sqrt{5^{2} \cdot 13}}{2 \cdot - 9}$

Step 6.4.1.5

Pull terms out from under the radical.

$v = \frac{- 1 \pm 5 \sqrt{13}}{2 \cdot - 9}$

Step 6.4.2

Multiply $2$ by $- 9$ .

$v = \frac{- 1 \pm 5 \sqrt{13}}{- 18}$

Step 6.4.3

Simplify $\frac{- 1 \pm 5 \sqrt{13}}{- 18}$ .

$v = \frac{1 \pm 5 \sqrt{13}}{18}$

Step 6.5

The final answer is the combination of both solutions.

$v = \frac{1 + 5 \sqrt{13}}{18}, \frac{1 - 5 \sqrt{13}}{18}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = \frac{1 + 5 \sqrt{13}}{18}, \frac{1 - 5 \sqrt{13}}{18}$

Decimal Form:

$v = 1.05709757 \dots, - 0.94598646 \dots$