Pre-Algebra Examples

$65 = \frac{1}{2} \cdot (n (n - 3))$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (n (n - 3)) = 65$ .

$\frac{1}{2} \cdot (n (n - 3)) = 65$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (n (n - 3))) = 2 \cdot 65$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $2 (\frac{1}{2} \cdot (n (n - 3)))$ .

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Step 3.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (n \cdot n + n \cdot - 3)) = 2 \cdot 65$

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Multiply $n$ by $n$ .

$2 (\frac{1}{2} \cdot (n^{2} + n \cdot - 3)) = 2 \cdot 65$

Step 3.1.1.2.2

Move $- 3$ to the left of $n$ .

$2 (\frac{1}{2} \cdot (n^{2} - 3 \cdot n)) = 2 \cdot 65$

Step 3.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} n^{2} + \frac{1}{2} (- 3 n)) = 2 \cdot 65$

Step 3.1.1.4

Combine $\frac{1}{2}$ and $n^{2}$ .

$2 (\frac{n^{2}}{2} + \frac{1}{2} (- 3 n)) = 2 \cdot 65$

Step 3.1.1.5

Multiply $\frac{1}{2} (- 3 n)$ .

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Step 3.1.1.5.1

Combine $- 3$ and $\frac{1}{2}$ .

$2 (\frac{n^{2}}{2} + \frac{- 3}{2} n) = 2 \cdot 65$

Step 3.1.1.5.2

Combine $\frac{- 3}{2}$ and $n$ .

$2 (\frac{n^{2}}{2} + \frac{- 3 n}{2}) = 2 \cdot 65$

Step 3.1.1.6

Move the negative in front of the fraction.

$2 (\frac{n^{2}}{2} - \frac{3 n}{2}) = 2 \cdot 65$

Step 3.1.1.7

Apply the distributive property.

$2 \frac{n^{2}}{2} + 2 (- \frac{3 n}{2}) = 2 \cdot 65$

Step 3.1.1.8

Cancel the common factor of $2$ .

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Step 3.1.1.8.1

Cancel the common factor.

$2 \frac{n^{2}}{2} + 2 (- \frac{3 n}{2}) = 2 \cdot 65$

Step 3.1.1.8.2

Rewrite the expression.

$n^{2} + 2 (- \frac{3 n}{2}) = 2 \cdot 65$

Step 3.1.1.9

Cancel the common factor of $2$ .

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Step 3.1.1.9.1

Move the leading negative in $- \frac{3 n}{2}$ into the numerator.

$n^{2} + 2 \frac{- 3 n}{2} = 2 \cdot 65$

Step 3.1.1.9.2

Cancel the common factor.

$n^{2} + 2 \frac{- 3 n}{2} = 2 \cdot 65$

Step 3.1.1.9.3

Rewrite the expression.

$n^{2} - 3 n = 2 \cdot 65$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $65$ .

$n^{2} - 3 n = 130$

Step 4

Subtract $130$ from both sides of the equation.

$n^{2} - 3 n - 130 = 0$

Step 5

Factor $n^{2} - 3 n - 130$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 130$ and whose sum is $- 3$ .

$- 13, 10$

Step 5.2

Write the factored form using these integers.

$(n - 13) (n + 10) = 0$

Step 6

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

$n - 13 = 0$

$n + 10 = 0$

Step 7

Set $n - 13$ equal to $0$ and solve for $n$ .

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Step 7.1

Set $n - 13$ equal to $0$ .

$n - 13 = 0$

Step 7.2

Add $13$ to both sides of the equation.

$n = 13$

Step 8

Set $n + 10$ equal to $0$ and solve for $n$ .

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Step 8.1

Set $n + 10$ equal to $0$ .

$n + 10 = 0$

Step 8.2

Subtract $10$ from both sides of the equation.

$n = - 10$

Step 9

The final solution is all the values that make $(n - 13) (n + 10) = 0$ true.

$n = 13, - 10$