Question

(1/(x-2))+((x-6)/(3x^2-12))=(1/(2-x))-1

Answer 1

Ответ: (7x+6)\(3x^2 - 12)

Пошаговое объяснение:

Answer 2

To solve the given equation, let's simplify it step by step:

(1/(x-2)) + ((x-6)/(3x^2-12)) = (1/(2-x)) - 1

First, we'll find a common denominator for the fractions on both sides of the equation. The common denominator will be (x - 2)(2 - x) since it combines the two denominators (x - 2) and (2 - x):

((2 - x) + (x - 6)(x - 2)) / [(x - 2)(2 - x)] = (1/(2-x)) - 1

Now, let's simplify further:

[(2 - x) + (x^2 - 8x + 12)] / [(x - 2)(2 - x)] = (1/(2-x)) - 1

Next, simplify the numerator:

(2 - x + x^2 - 8x + 12) / [(x - 2)(2 - x)] = (1/(2-x)) - 1

Combine like terms in the numerator:

(x^2 - 9x + 14) / [(x - 2)(2 - x)] = (1/(2-x)) - 1

Notice that the denominator (x - 2)(2 - x) is equal to -(x - 2)(x - 2), so we can simplify further:

(x^2 - 9x + 14) / (-(x - 2)(x - 2)) = (1/(2-x)) - 1

Since the denominators are equal and non-zero, we can cancel them out:

x^2 - 9x + 14 = 1 - (2 - x)

Now, simplify the right side:

x^2 - 9x + 14 = 1 - 2 + x

Combine like terms:

x^2 - 9x + 14 = -1 + x

Rearrange the equation to bring all terms to one side:

x^2 - 9x - x + 14 + 1 = 0

x^2 - 10x + 15 = 0

Now, let's factorize the quadratic equation:

(x - 5)(x - 3) = 0

Setting each factor to zero:

x - 5 = 0 or x - 3 = 0

Solving for x:

x = 5 or x = 3

Therefore, the solutions to the equation are x = 5 and x = 3.