Question

solve: \\(\frac{x^2 - x - 6}{x^2} = \frac{x - 6}{2x} + \frac{2x + 12}{x}\\) after multiplying each side of the equation by the lcd and simplifying, the resulting equation is

Explanation:

Step1: Identify the LCD

The denominators are $x^2$, $2x$, and $x$. The least common denominator (LCD) is $2x^2$.

Step2: Multiply all terms by LCD

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

Step3: Simplify each term

Simplify left term: $2(x^2 - x - 6) = 2x^2 - 2x - 12$
Simplify middle term: $x(x - 6) = x^2 - 6x$
Simplify right term: $2x(2x + 12) = 4x^2 + 24x$
Resulting equation after simplification:
$$2x^2 - 2x - 12 = x^2 - 6x + 4x^2 + 24x$$

Step4: Combine like terms on right

$$2x^2 - 2x - 12 = 5x^2 + 18x$$

Step5: Rearrange to standard form

Subtract $2x^2 - 2x - 12$ from both sides:
$$0 = 3x^2 + 20x + 12$$
Or rewritten as:
$$3x^2 + 20x + 12 = 0$$

Answer:

$3x^2 + 20x + 12 = 0$