Basic Math Examples

$3 n \cdot \frac{n + 3}{2} = 756$

Step 1

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 n (n + 3)}{2} = \frac{2}{3} \cdot 756$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Cancel the common factor of $2$ .

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Step 2.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 n (n + 3)}{2} = \frac{2}{3} \cdot 756$

Step 2.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 n (n + 3)) = \frac{2}{3} \cdot 756$

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Factor $3$ out of $3 n (n + 3)$ .

$\frac{1}{3} (3 (n (n + 3))) = \frac{2}{3} \cdot 756$

Step 2.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 (n (n + 3))) = \frac{2}{3} \cdot 756$

Step 2.1.1.2.3

Rewrite the expression.

$n (n + 3) = \frac{2}{3} \cdot 756$

Step 2.1.1.3

Apply the distributive property.

$n \cdot n + n \cdot 3 = \frac{2}{3} \cdot 756$

Step 2.1.1.4

Simplify the expression.

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Step 2.1.1.4.1

Multiply $n$ by $n$ .

$n^{2} + n \cdot 3 = \frac{2}{3} \cdot 756$

Step 2.1.1.4.2

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

$n^{2} + 3 n = \frac{2}{3} \cdot 756$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{2}{3} \cdot 756$ .

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Factor $3$ out of $756$ .

$n^{2} + 3 n = \frac{2}{3} \cdot (3 (252))$

Step 2.2.1.1.2

Cancel the common factor.

$n^{2} + 3 n = \frac{2}{3} \cdot (3 \cdot 252)$

Step 2.2.1.1.3

Rewrite the expression.

$n^{2} + 3 n = 2 \cdot 252$

Step 2.2.1.2

Multiply $2$ by $252$ .

$n^{2} + 3 n = 504$

Step 3

Subtract $504$ from both sides of the equation.

$n^{2} + 3 n - 504 = 0$

Step 4

Factor $n^{2} + 3 n - 504$ using the AC method.

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Step 4.1

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 $- 504$ and whose sum is $3$ .

$- 21, 24$

Step 4.2

Write the factored form using these integers.

$(n - 21) (n + 24) = 0$

Step 5

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

$n - 21 = 0$

$n + 24 = 0$

Step 6

Set $n - 21$ equal to $0$ and solve for $n$ .

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Step 6.1

Set $n - 21$ equal to $0$ .

$n - 21 = 0$

Step 6.2

Add $21$ to both sides of the equation.

$n = 21$

Step 7

Set $n + 24$ equal to $0$ and solve for $n$ .

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Step 7.1

Set $n + 24$ equal to $0$ .

$n + 24 = 0$

Step 7.2

Subtract $24$ from both sides of the equation.

$n = - 24$

Step 8

The final solution is all the values that make $(n - 21) (n + 24) = 0$ true.

$n = 21, - 24$