Question

(х-2 дробь х^2-х) +(1 дробь х^2+х)=(2 дробь х^2-1)

Answer 1

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Answer 2

To solve the equation, we need to find the value of 'x' that satisfies the equation:

$\frac{x-2}{x^2-x} + \frac{1}{x^2+x} = \frac{2}{x^2-1}$

To simplify this equation, let's first find a common denominator for all the fractions. The common denominator is $(x^2 - x)(x^2 + x)(x^2 - 1)$. Now, let's rewrite each fraction with the common denominator:

$\frac{(x-2)(x^2 + x)(x^2 - 1)}{(x^2-x)(x^2 + x)(x^2 - 1)} + \frac{(x^2 - x)(x^2 - 1)}{(x^2-x)(x^2 + x)(x^2 - 1)} = \frac{2(x^2-x)(x^2 + x)}{(x^2-x)(x^2 + x)(x^2 - 1)}$

Next, we can combine the fractions on the left side:

$\frac{(x-2)(x^2 + x)(x^2 - 1) + (x^2 - x)(x^2 - 1)}{(x^2-x)(x^2 + x)(x^2 - 1)} = \frac{2(x^2-x)(x^2 + x)}{(x^2-x)(x^2 + x)(x^2 - 1)}$

Now, let's cancel out the common terms in the numerator and denominator:

$(x-2)(x^2 + x)(x^2 - 1) + (x^2 - x)(x^2 - 1) = 2(x^2 - x)(x^2 + x)$

Expand the expressions:

$x^5 - 3x^4 + 2x^3 + 4x^2 - 2x - 2x^4 + 2x^3 + x^2 - x = 2x^4 - 2x^3 + 2x^3 - 2x^2$

Combine like terms:

$x^5 - 5x^4 + 3x^3 + 5x^2 - 3x = 0$

Now, we have a fifth-degree polynomial equation. Unfortunately, it's not straightforward to solve this equation analytically, and we would need to use numerical methods to find approximate solutions for 'x'.

If you have a specific value of 'x' in mind, you can substitute it into the original equation to check if it satisfies the equation. However, finding all possible solutions to this equation requires more advanced numerical techniques.