Finite Math Examples

$2 v^{2} - 10 v - 20 = 8$

Step 1

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

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Step 1.1

Add $20$ to both sides of the equation.

$2 v^{2} - 10 v = 8 + 20$

Step 1.2

Add $8$ and $20$ .

$2 v^{2} - 10 v = 28$

Step 2

Divide each term in $2 v^{2} - 10 v = 28$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 v^{2} - 10 v = 28$ by $2$ .

$\frac{2 v^{2}}{2} + \frac{- 10 v}{2} = \frac{28}{2}$

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

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{2 v^{2}}{2} + \frac{- 10 v}{2} = \frac{28}{2}$

Step 2.2.1.1.2

Divide $v^{2}$ by $1$ .

$v^{2} + \frac{- 10 v}{2} = \frac{28}{2}$

Step 2.2.1.2

Cancel the common factor of $- 10$ and $2$ .

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Step 2.2.1.2.1

Factor $2$ out of $- 10 v$ .

$v^{2} + \frac{2 (- 5 v)}{2} = \frac{28}{2}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $2$ out of $2$ .

$v^{2} + \frac{2 (- 5 v)}{2 (1)} = \frac{28}{2}$

Step 2.2.1.2.2.2

Cancel the common factor.

$v^{2} + \frac{2 (- 5 v)}{2 \cdot 1} = \frac{28}{2}$

Step 2.2.1.2.2.3

Rewrite the expression.

$v^{2} + \frac{- 5 v}{1} = \frac{28}{2}$

Step 2.2.1.2.2.4

Divide $- 5 v$ by $1$ .

$v^{2} - 5 v = \frac{28}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $28$ by $2$ .

$v^{2} - 5 v = 14$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{5}{2})}^{2}$

Step 4

Add the term to each side of the equation.

$v^{2} - 5 v + {(- \frac{5}{2})}^{2} = 14 + {(- \frac{5}{2})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$v^{2} - 5 v + {(- 1)}^{2} {(\frac{5}{2})}^{2} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$v^{2} - 5 v + {(- 1)}^{2} \frac{5^{2}}{2^{2}} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

$v^{2} - 5 v + 1 \frac{5^{2}}{2^{2}} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.1.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$v^{2} - 5 v + \frac{5^{2}}{2^{2}} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.1.1.4

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

$v^{2} - 5 v + \frac{25}{2^{2}} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.1.1.5

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

$v^{2} - 5 v + \frac{25}{4} = 14 + {(- \frac{5}{2})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $14 + {(- \frac{5}{2})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$v^{2} - 5 v + \frac{25}{4} = 14 + {(- 1)}^{2} {(\frac{5}{2})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$v^{2} - 5 v + \frac{25}{4} = 14 + {(- 1)}^{2} \frac{5^{2}}{2^{2}}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$v^{2} - 5 v + \frac{25}{4} = 14 + 1 \frac{5^{2}}{2^{2}}$

Step 5.2.1.1.3

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$v^{2} - 5 v + \frac{25}{4} = 14 + \frac{5^{2}}{2^{2}}$

Step 5.2.1.1.4

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

$v^{2} - 5 v + \frac{25}{4} = 14 + \frac{25}{2^{2}}$

Step 5.2.1.1.5

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

$v^{2} - 5 v + \frac{25}{4} = 14 + \frac{25}{4}$

Step 5.2.1.2

To write $14$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$v^{2} - 5 v + \frac{25}{4} = 14 \cdot \frac{4}{4} + \frac{25}{4}$

Step 5.2.1.3

Combine $14$ and $\frac{4}{4}$ .

$v^{2} - 5 v + \frac{25}{4} = \frac{14 \cdot 4}{4} + \frac{25}{4}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$v^{2} - 5 v + \frac{25}{4} = \frac{14 \cdot 4 + 25}{4}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $14$ by $4$ .

$v^{2} - 5 v + \frac{25}{4} = \frac{56 + 25}{4}$

Step 5.2.1.5.2

Add $56$ and $25$ .

$v^{2} - 5 v + \frac{25}{4} = \frac{81}{4}$

Step 6

Factor the perfect trinomial square into ${(v - \frac{5}{2})}^{2}$ .

${(v - \frac{5}{2})}^{2} = \frac{81}{4}$

Step 7

Solve the equation for $v$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$v - \frac{5}{2} = \pm \sqrt{\frac{81}{4}}$

Step 7.2

Simplify $\pm \sqrt{\frac{81}{4}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{81}{4}}$ as $\frac{\sqrt{81}}{\sqrt{4}}$ .

$v - \frac{5}{2} = \pm \frac{\sqrt{81}}{\sqrt{4}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $81$ as $9^{2}$ .

$v - \frac{5}{2} = \pm \frac{\sqrt{9^{2}}}{\sqrt{4}}$

Step 7.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$v - \frac{5}{2} = \pm \frac{9}{\sqrt{4}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$v - \frac{5}{2} = \pm \frac{9}{2}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$v - \frac{5}{2} = \frac{9}{2}$

Step 7.3.2

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

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Step 7.3.2.1

Add $\frac{5}{2}$ to both sides of the equation.

$v = \frac{9}{2} + \frac{5}{2}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$v = \frac{9 + 5}{2}$

Step 7.3.2.3

Add $9$ and $5$ .

$v = \frac{14}{2}$

Step 7.3.2.4

Divide $14$ by $2$ .

$v = 7$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$v - \frac{5}{2} = - \frac{9}{2}$

Step 7.3.4

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

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Step 7.3.4.1

Add $\frac{5}{2}$ to both sides of the equation.

$v = - \frac{9}{2} + \frac{5}{2}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$v = \frac{- 9 + 5}{2}$

Step 7.3.4.3

Add $- 9$ and $5$ .

$v = \frac{- 4}{2}$

Step 7.3.4.4

Divide $- 4$ by $2$ .

$v = - 2$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$v = 7, - 2$