Question

3/x - 6/ x(x+2) = 8/ x^2 - 4

Answer 1

Ответ:

x= 4 2/3

Объяснение:

3/х-6/x(x+2)=8/(x²-4) по формуле сокр. умножения

3/x-6/x(x+2)=8/(x-2)(x+2)

3/x-6/x(x+2)-8/(x-2)(x+2)=0;

3*(x-2)(x+2)/x(x-2)(x+2)-6(x-2)/x(x-2)(x+2) - 8x/x(x-2)(x+2)=(3(x²-4)-6(x-2)-8x)/x(x²-4)=

=(3x²-12-6x+12-8x)/(x³-4x)= (3x²-14x)/(x³-4x) =x(3x-14)/x(x²-4)=(3x-14)/(x²-4)

(3x-14)/(x²-4)=0:

3x-14=0*(x²-4);

3x-14=0;

3x=14;

x=14/3;

x=4 2/3.

Answer 2

To solve the equation 3/x - 6/(x(x+2)) = 8/(x^2 - 4), we need to follow these steps:

Step 1: Factorize the denominators. Step 2: Find a common denominator for the fractions. Step 3: Simplify the equation. Step 4: Solve for x.

Let's go through each step:

Step 1: Factorize the denominators. x(x+2) can't be simplified further, but x^2 - 4 can be factored using the difference of squares formula: x^2 - 4 = (x+2)(x-2).

Step 2: Find a common denominator for the fractions. The common denominator will be x(x+2)(x-2).

Step 3: Simplify the equation. To rewrite the fractions with the common denominator, we need to multiply each fraction by the necessary factor to get the common denominator in the denominator. The equation becomes:

(3 * (x+2)(x-2))/(x(x+2)(x-2)) - (6 * x-2)/(x(x+2)(x-2)) = (8 * x)/(x(x+2)(x-2))

Step 4: Solve for x. Now that we have a common denominator, we can combine the numerators on the left side:

(3(x+2)(x-2) - 6(x-2))/(x(x+2)(x-2)) = (8x)/(x(x+2)(x-2))

Expand the numerators:

(3x^2 - 6 - 6x + 12)/(x(x+2)(x-2)) = (8x)/(x(x+2)(x-2))

Combine like terms:

(3x^2 - 6x + 6)/(x(x+2)(x-2)) = (8x)/(x(x+2)(x-2))

Now, cross-multiply:

(3x^2 - 6x + 6) * x = 8x * (x+2)(x-2)

Expand and simplify:

3x^3 - 6x^2 + 6x = 8x * (x^2 - 4)

3x^3 - 6x^2 + 6x = 8x^3 - 32x

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

3x^3 - 6x^2 + 6x - 8x^3 + 32x = 0

Combine like terms:

-5x^3 - 6x^2 + 38x = 0

Now, this is a cubic equation, and solving it for x can be a bit more complex than solving linear or quadratic equations. However, you can use numerical methods or calculators to find approximate solutions for x.

Alternatively, if you had any specific values for the equation, I could help you calculate the solutions for those particular values.