Algebra Examples

$\frac{50}{4 x - 12} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1

Simplify $\frac{50}{4 x - 12} + \frac{x - 4}{x^{2} + x - 12}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Cancel the common factor of $50$ and $4 x - 12$ .

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Step 1.1.1.1

Factor $2$ out of $50$ .

$\frac{2 (25)}{4 x - 12} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.1.2

Cancel the common factors.

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Step 1.1.1.2.1

Factor $2$ out of $4 x$ .

$\frac{2 (25)}{2 (2 x) - 12} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.1.2.2

Factor $2$ out of $- 12$ .

$\frac{2 \cdot 25}{2 (2 x) + 2 \cdot - 6} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.1.2.3

Factor $2$ out of $2 (2 x) + 2 (- 6)$ .

$\frac{2 (25)}{2 (2 x - 6)} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.1.2.4

Cancel the common factor.

$\frac{2 \cdot 25}{2 (2 x - 6)} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.1.2.5

Rewrite the expression.

$\frac{25}{2 x - 6} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.2

Factor $2$ out of $2 x - 6$ .

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Step 1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{25}{2 (x) - 6} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.2.2

Factor $2$ out of $- 6$ .

$\frac{25}{2 x + 2 \cdot - 3} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.2.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{25}{2 (x - 3)} + \frac{x - 4}{x^{2} + x - 12} = \frac{x}{2}$

Step 1.1.3

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

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Step 1.1.3.1

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 1.1.3.2

Write the factored form using these integers.

$\frac{25}{2 (x - 3)} + \frac{x - 4}{(x - 3) (x + 4)} = \frac{x}{2}$

Step 1.2

To write $\frac{25}{2 (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{25}{2 (x - 3)} \cdot \frac{x + 4}{x + 4} + \frac{x - 4}{(x - 3) (x + 4)} = \frac{x}{2}$

Step 1.3

To write $\frac{x - 4}{(x - 3) (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{25}{2 (x - 3)} \cdot \frac{x + 4}{x + 4} + \frac{x - 4}{(x - 3) (x + 4)} \cdot \frac{2}{2} = \frac{x}{2}$

Step 1.4

Write each expression with a common denominator of $2 (x - 3) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{25}{2 (x - 3)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{25 (x + 4)}{2 (x - 3) (x + 4)} + \frac{x - 4}{(x - 3) (x + 4)} \cdot \frac{2}{2} = \frac{x}{2}$

Step 1.4.2

Multiply $\frac{x - 4}{(x - 3) (x + 4)}$ by $\frac{2}{2}$ .

$\frac{25 (x + 4)}{2 (x - 3) (x + 4)} + \frac{(x - 4) \cdot 2}{(x - 3) (x + 4) \cdot 2} = \frac{x}{2}$

Step 1.4.3

Reorder the factors of $2 (x - 3) (x + 4)$ .

$\frac{25 (x + 4)}{2 (x + 4) (x - 3)} + \frac{(x - 4) \cdot 2}{(x - 3) (x + 4) \cdot 2} = \frac{x}{2}$

Step 1.4.4

Reorder the factors of $(x - 3) (x + 4) \cdot 2$ .

$\frac{25 (x + 4)}{2 (x + 4) (x - 3)} + \frac{(x - 4) \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{25 (x + 4) + (x - 4) \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Apply the distributive property.

$\frac{25 x + 25 \cdot 4 + (x - 4) \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.2

Multiply $25$ by $4$ .

$\frac{25 x + 100 + (x - 4) \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.3

Apply the distributive property.

$\frac{25 x + 100 + x \cdot 2 - 4 \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.4

Move $2$ to the left of $x$ .

$\frac{25 x + 100 + 2 \cdot x - 4 \cdot 2}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.5

Multiply $- 4$ by $2$ .

$\frac{25 x + 100 + 2 \cdot x - 8}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.6

Add $25 x$ and $2 x$ .

$\frac{27 x + 100 - 8}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 1.6.7

Subtract $8$ from $100$ .

$\frac{27 x + 92}{2 (x + 4) (x - 3)} = \frac{x}{2}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 5.10894399, - 2.660238, 6.769182$

Step 3