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x^2-x-6/x^2=x-6/2x+2x+12/x


After multiplying each side by the equation LCD and simplifying, the resulting equation is...

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morgeron

first we look at the equation an determine our common denominator.

(X²-X-6) / X² = (X-6) / 2X + (2X+12) / X    (let's use 2X² as the comm denom)

multiply the first fraction by 2, the second by X, and the third by 2x

(2x²-2x-12) / 2x² = (x²-6x) / 2x² + (4x²+24x) / 2x²  (dropping the denominator)

2x² - 2x - 12 = x² - 6x + 4x² + 24x      (simplify)

-3x² - 20x - 12  (multiply by -1)
3x² + 20x + 12
(x + 6) (3x + 2)    (factored version)

MrRoyal

Equivalent equations are equations with equal values

The solution is: \(x = -\frac 23\) or \(x = -6\)

The equation is given as:

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

Take LCM

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

\(\frac{x^2-x-6}{x^2 }= \frac{x-6 + 4x + 24}{2x}\)

Multiply both sides by x

\(\frac{x^2-x-6}{x }= \frac{x-6 + 4x + 24}{2}\)

\(\frac{x^2-x-6}{x }= \frac{5x+18}{2}\)

Cross multiply

\(2x^2 - 2x - 12 = 5x^2 + 18x\)

Collect like terms

\(5x^2 -2x^2 + 18x + 2x + 12 = 0\)

\(3x^2 + 20x + 12 = 0\)

Expand

\(3x^2 + 18x + 2x + 12 = 0\)

Factorize

\(3x(x + 6) + 2(x + 6) = 0\)

Factor out x + 6

\((3x + 2)(x + 6) = 0\)

Split

\((3x + 2)= 0\) or \((x + 6) = 0\)

Solve for x

\(x = -\frac 23\) or \(x = -6\)

Read more about equivalent equations at:

https://brainly.com/question/19521931

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