1 + 2(17-6x)/х2-6х+8 = 1-2х/х-4 - 11/х-2

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вопрос

To solve this equation, we need to simplify and then solve for x.

Starting with the given equation:
1 + 2(17-6x)/(x^2-6x+8) = 1 - 2x/(x-4) - 11/(x-2)

First, simplify the expressions on both sides:
1 + 2(17-6x)/(x^2-6x+8) = 1 - 2x/(x-4) - 11/(x-2)

1 + 2(17-6x)/(x^2-6x+8) = 1 - 2x/(x-4) - 11/(x-2)

1 + 2(17-6x)/(x-4)(x-2) = 1 - 2x/(x-4) - 11/(x-2)

1 + 2(17-6x)/(x-4)(x-2) = 1 - 2x/(x-4) - 11/(x-2)

Now, get a common denominator:
1 + 2(17-6x)/(x-4)(x-2) = (x-4 - 2x - 11)/((x-4)(x-2))

1 + 2(17-6x)/(x-4)(x-2) = (-x - 15)/((x-4)(x-2))

Now, combine like terms:
1 + 2(17-6x)/(x-4)(x-2) = (-x - 15)/((x-4)(x-2))

1 + (34 - 12x)/(x-4)(x-2) = (-x - 15)/((x-4)(x-2))

Now, cross multiply:
(x-4)(x-2)(1 + (34 - 12x)/(x-4)(x-2)) = (-x - 15)

(x-4)(x-2) + (34 - 12x) = (-x - 15)

Expand and simplify:
(x^2 - 6x + 8) + 34 - 12x = -x - 15

x^2 - 6x + 8 + 34 - 12x = -x - 15

x^2 - 6x - 12x + 42 = -x - 15

x^2 - 18x + 42 = -x - 15

Rearrange the equation to set it equal to zero:
x^2 - 17x + 57 = 0

Now, we can solve for x using the quadratic formula:
x = [17 ± sqrt((-17)^2 - 4(1)(57))] / 2(1)

x = [17 ± sqrt(289 - 228)] / 2

x = [17 ± sqrt(61)] / 2

Therefore, the solutions for x are:
x = (17 + sqrt(61)) / 2 or x = (17 - sqrt(61)) / 2