Question

X-1/x+2 +x+2/x-2 = 16/x²-4

Answer 1

Ответ: х = 2; х = -2.5

......

Answer 2

To solve the given equation, we'll first find a common denominator for all the fractions and then combine like terms. Let's begin:

Given equation: (X-1)/(X+2) + (X+2)/(X-2) = 16/(X^2-4)

Step 1: Find a common denominator for all the fractions. The common denominator for the fractions on the left side is (X+2)(X-2) since (X^2-4) can be factored as (X+2)(X-2). So, let's rewrite the fractions using the common denominator:

[(X-1)(X-2) + (X+2)(X+2)] / [(X+2)(X-2)] = 16 / (X^2-4)

Step 2: Expand and simplify the numerator on the left side:

[X^2 - 3X + 2 + X^2 + 4X + 4] / [(X+2)(X-2)] = 16 / (X^2-4)

Step 3: Combine like terms in the numerator:

[2X^2 + X + 6] / [(X+2)(X-2)] = 16 / (X^2-4)

Step 4: Cross-multiply to eliminate the denominators:

(2X^2 + X + 6) * (X^2-4) = 16 * [(X+2)(X-2)]

Step 5: Expand and simplify both sides of the equation:

2X^4 - 8X^2 + X^3 - 4X + 6X^2 - 24 = 16 * (X^2 - 4)

Step 6: Combine like terms:

2X^4 + X^3 - 2X^2 - 4X - 24 = 16X^2 - 64

Step 7: Move all the terms to one side to set the equation to zero:

2X^4 + X^3 - 2X^2 - 4X - 16X^2 + 64 - 24 = 0

2X^4 + X^3 - 18X^2 - 4X + 40 = 0

Now, this is a quartic equation, and solving it for X requires more advanced techniques like numerical methods or factoring if possible. However, it's important to double-check the equation to ensure there are no typos or errors in the original problem statement.

If this is indeed the correct equation, you may use numerical methods or a graphing calculator to find the approximate solutions for X.