Algebra Examples

$\frac{n}{n - 4} + n = \frac{12 - 4 n}{n - 4}$

Step 1

Subtract $\frac{12 - 4 n}{n - 4}$ from both sides of the equation.

$\frac{n}{n - 4} + n - \frac{12 - 4 n}{n - 4} = 0$

Step 2

Simplify $\frac{n}{n - 4} + n - \frac{12 - 4 n}{n - 4}$ .

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Step 2.1

Combine the numerators over the common denominator.

$n + \frac{n - (12 - 4 n)}{n - 4} = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the distributive property.

$n + \frac{n - 1 \cdot 12 - (- 4 n)}{n - 4} = 0$

Step 2.2.2

Multiply $- 1$ by $12$ .

$n + \frac{n - 12 - (- 4 n)}{n - 4} = 0$

Step 2.2.3

Multiply $- 4$ by $- 1$ .

$n + \frac{n - 12 + 4 n}{n - 4} = 0$

Step 2.3

Add $n$ and $4 n$ .

$n + \frac{5 n - 12}{n - 4} = 0$

Step 2.4

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

$\frac{n (n - 4)}{n - 4} + \frac{5 n - 12}{n - 4} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{n (n - 4) + 5 n - 12}{n - 4} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{n \cdot n + n \cdot - 4 + 5 n - 12}{n - 4} = 0$

Step 2.6.2

Multiply $n$ by $n$ .

$\frac{n^{2} + n \cdot - 4 + 5 n - 12}{n - 4} = 0$

Step 2.6.3

Move $- 4$ to the left of $n$ .

$\frac{n^{2} - 4 \cdot n + 5 n - 12}{n - 4} = 0$

Step 2.6.4

Add $- 4 n$ and $5 n$ .

$\frac{n^{2} + n - 12}{n - 4} = 0$

Step 2.6.5

Factor $n^{2} + n - 12$ using the AC method.

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Step 2.6.5.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.6.5.2

Write the factored form using these integers.

$\frac{(n - 3) (n + 4)}{n - 4} = 0$

Step 3

Set the numerator equal to zero.

$(n - 3) (n + 4) = 0$

Step 4

Solve the equation for $n$ .

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Step 4.1

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

$n - 3 = 0$

$n + 4 = 0$

Step 4.2

Set $n - 3$ equal to $0$ and solve for $n$ .

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Step 4.2.1

Set $n - 3$ equal to $0$ .

$n - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$n = 3$

Step 4.3

Set $n + 4$ equal to $0$ and solve for $n$ .

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Step 4.3.1

Set $n + 4$ equal to $0$ .

$n + 4 = 0$

Step 4.3.2

Subtract $4$ from both sides of the equation.

$n = - 4$

Step 4.4

The final solution is all the values that make $(n - 3) (n + 4) = 0$ true.

$n = 3, - 4$