Question

Х/(х-2)+8/(4-х^2)=1/(х+2)​

Answer 1

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Answer 2

To solve the given equation, we'll first simplify and manipulate it to make it easier to work with. Then, we'll find the values of 'x' that satisfy the equation. Let's start step-by-step:

Step 1: Factorize the denominator in the second term. The denominator in the second term is a difference of squares, which can be factorized as follows: 4 - x^2 = (2 + x)(2 - x)

Step 2: Find a common denominator for the terms on the left side. The common denominator for the terms on the left side is (x - 2)(2 + x)(2 - x).

Step 3: Rewrite the fractions with the common denominator. Now, rewrite the equation as follows: [(x - 2)(2 + x)(2 - x)/(x - 2)] + [8(x - 2)(2 + x)/(4 - x^2)] = [(x - 2)(2 + x)(2 - x)/(x + 2)]

Step 4: Eliminate the common factor (x - 2) from both sides of the equation. After canceling the common factor (x - 2) from both sides, we get: (2 + x)(2 - x) + 8(x + 2) = (x + 2)(2 + x)(2 - x)

Step 5: Expand and simplify the expressions. Expand both sides of the equation: (2^2 - x^2) + 8(x + 2) = (2^2 - x^2)(x + 2)

Simplify: 4 - x^2 + 8x + 16 = 4x + 4 - x^2

Step 6: Bring like terms to one side of the equation. Subtract (4x + 4 - x^2) from both sides: 4 - x^2 + 8x + 16 - (4x + 4 - x^2) = 0

Simplify: 4 - x^2 + 8x + 16 - 4x - 4 + x^2 = 0

Step 7: Simplify further and combine like terms. On the left side, x^2 and -x^2 cancel out, and 8x - 4x = 4x: 4x + 16 - 4 = 0

Step 8: Solve for 'x'. 4x + 12 = 0 4x = -12 x = -12/4 x = -3

Step 9: Check for extraneous solutions. Now, we must check if x = -3 is an extraneous solution, meaning it causes any of the original denominators to be zero, which would result in division by zero (undefined). Let's check each denominator.

For (x - 2), when x = -3: (-3 - 2) = -5 ≠ 0

For (4 - x^2), when x = -3: (4 - (-3)^2) = (4 - 9) = -5 ≠ 0

For (x + 2), when x = -3: (-3 + 2) = -1 ≠ 0

Since none of the denominators become zero when x = -3, the solution is valid.

Thus, the solution to the equation is x = -3.