Question

(x + 4)/(x^2 - 4*x) - 1/x = 2 - (x - 6)/(x - 4)

Answer 1

Ответ на фотографии!

Answer 2

To solve the given equation $\frac{x + 4}{x^2 - 4x} - \frac{1}{x} = 2 - \frac{x - 6}{x - 4}$, let's find a common denominator and simplify the equation.

The common denominator for the fractions on the left side can be $x(x - 4)$, so let's rewrite the equation with this common denominator:

$$\frac{x + 4}{x^2 - 4x} - \frac{1}{x} = \frac{(x + 4)(x)}{x(x - 4)} - \frac{(x - 4)}{x(x - 4)} = \frac{x(x + 4) - (x - 4)}{x(x - 4)}$$

Now, the equation becomes:

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

Combine the fractions on the left side:

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

Simplify the numerator:

$$\frac{x^2 + 3x + 4}{x(x - 4)} = 2 - \frac{x - 6}{x - 4}$$

Now, find a common denominator for the right side, which is $x - 4$:

$$\frac{x^2 + 3x + 4}{x(x - 4)} = \frac{2(x - 4) - (x - 6)}{x - 4}$$

Simplify the right side:

$$\frac{x^2 + 3x + 4}{x(x - 4)} = \frac{2x - 8 - x + 6}{x - 4}$$

Combine like terms on the right side:

$$\frac{x^2 + 3x + 4}{x(x - 4)} = \frac{x - 2}{x - 4}$$

Now, we have the equation:

$$\frac{x^2 + 3x + 4}{x(x - 4)} = \frac{x - 2}{x - 4}$$

To solve for $x$, we can cross-multiply:

$$(x^2 + 3x + 4)(x - 4) = (x - 2)x$$

Expand and simplify the equation:

$$x^3 - 4x^2 + 3x^2 + 12x - 4x - 16 = x^2 - 2x$$

Combine like terms:

$$x^3 - x^2 + 16x - 16 = x^2 - 2x$$

Move all terms to one side to set the equation to zero:

$$x^3 - 2x^2 + 18x - 16 = 0$$

This is the cubic equation that needs to be solved for $x$. The solutions to this cubic equation can be found using various methods, such as factoring, synthetic division, or numerical methods. The solutions will be the values of $x$ that satisfy the original equation.