Question

3х/х^3-1 - 5/4х^2+4х+4 - 1/2(1-х)=0

Answer 1

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Answer 2

To solve the equation, we need to find the values of "x" that make the equation true. Let's first simplify the given equation step by step:

1. Combine all the fractions on the left side of the equation: (3x / (x^3 - 1)) - (5 / (4x^2 + 4x + 4)) - (1 / 2(1 - x)) = 0

2. The first fraction can be further simplified: 3x / (x^3 - 1) = 3x / (x^3 - 1)

3. The second fraction can be simplified by factoring out 4 from the denominator: 5 / (4x^2 + 4x + 4) = 5 / 4(x^2 + x + 1)

4. The third fraction can be simplified: 1 / 2(1 - x) = 1 / (2 - 2x)

Now, the equation becomes: 3x / (x^3 - 1) - 5 / 4(x^2 + x + 1) - 1 / (2 - 2x) = 0

Next, we need to find a common denominator for all the fractions. The common denominator is (4(x^2 + x + 1)(2 - 2x)(x^3 - 1)). Multiply each term by this common denominator to eliminate the fractions:

(4(x^2 + x + 1)(2 - 2x)(x^3 - 1))(3x / (x^3 - 1) - 5 / 4(x^2 + x + 1) - 1 / (2 - 2x)) = 0

Simplifying further:

4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(3x) - 4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(5 / 4(x^2 + x + 1)) - 4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(1 / (2 - 2x)) = 0

Now, we can cancel out some common terms:

4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(3x) - 4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(5 / 4) - 4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(1 / (2 - 2x)) = 0

Simplifying further:

4(x^2 + x + 1)(2 - 2x)(x^3 - 1)(3x) - 5(x^2 + x + 1)(2 - 2x)(x^3 - 1) - 2(x^2 + x + 1)(2 - 2x)(x^3 - 1)(1 / (2 - 2x)) = 0

Now, let's factor out the common terms:

(2 - 2x)(x^3 - 1)(4(x^2 + x + 1)(3x) - 5(x^2 + x + 1) - 2(1 / (2 - 2x))) = 0

Now, we have factored the equation. To find the values of "x," we need to solve each part of the equation for zero:

1. (2 - 2x) = 0 2 - 2x = 0 -2x = -2 x = -2 / -2 x = 1

2. (x^3 - 1) = 0 x^3 - 1 = 0 x^3 = 1 x = 1^(1/3) x = 1

3. (4(x^2 + x + 1)(3x) - 5(x^2 + x + 1) - 2(1 / (2 - 2x))) = 0 Since we already found that x = 1 satisfies this equation, let's substitute x = 1 into the equation: (4(1^2 + 1 + 1)(3 * 1) - 5(1^2 + 1 + 1) - 2(1 / (2 - 2 * 1))) = 0 (4(3) - 5(3) - 2(1 / 0)) = 0 (12 - 15 - 2(∞)) = 0

Here, we run into an issue because we have a division by zero in the equation. Therefore, x = 1 is the only valid solution for this equation.

So, the solution to the equation is x = 1.