Question

X/x-1+x+1/x+3=1/x^2+2x-3

Answer 1

x(x + 3)(x - 1) = x²(x + 2),

x(x² - x + 3x - 3) = x³ + 2x²,

х(х² + 2х - 3) - х³ - 2х² = 0,

x³ + 2x² - 3x - x³ - 2x² = 0,

-3x = 0,

x = 0.

Ответ: 0.

Answer 2

It seems like you've presented an equation:

X/(x-1) + x + 1/(x+3) = 1/(x^2 + 2x - 3)

If you're looking for a solution to this equation, I can guide you through the steps to solve it.

Step 1: Factor the denominator of the right-hand side The denominator on the right-hand side can be factored: x^2 + 2x - 3 = (x + 3)(x - 1).

Step 2: Find a common denominator To combine the fractions on the left-hand side, you need a common denominator. The common denominator for the fractions X/(x-1) and 1/(x+3) would be (x-1)(x+3).

Step 3: Rewrite fractions with the common denominator Rewrite the fractions with the common denominator:

X/(x-1) = X(x+3)/[(x-1)(x+3)] 1/(x+3) = (x-1)/[(x-1)(x+3)]

Now the equation becomes:

X(x+3)/[(x-1)(x+3)] + x + (x-1)/[(x-1)(x+3)] = 1/[(x+3)(x-1)]

Step 4: Combine the fractions Combine the fractions on the left-hand side:

[X(x+3) + (x^2 - 1)] / [(x-1)(x+3)] + x = 1/[(x+3)(x-1)]

Simplify:

X(x+3) + x^2 - 1 / [(x-1)(x+3)] + x = 1/[(x+3)(x-1)]

Step 5: Further simplify the left-hand side Combine the terms involving X:

X(x+3) + x^2 - 1 = Xx + 3X + x^2 - 1

Now the equation becomes:

(Xx + 3X + x^2 - 1) / [(x-1)(x+3)] + x = 1/[(x+3)(x-1)]

Step 6: Combine the fractions again Combine the fractions on the left-hand side:

[Xx + 3X + x^2 - 1 + x(x-1)(x+3)] / [(x-1)(x+3)] = 1/[(x+3)(x-1)]

Simplify:

[Xx + 3X + x^2 - 1 + x(x^2 + 2x - 3)] / [(x-1)(x+3)] = 1/[(x+3)(x-1)]

Step 7: Simplify the right-hand side Simplify the expression on the right-hand side:

[Xx + 3X + x^2 - 1 + x^3 + 2x^2 - 3x] / [(x-1)(x+3)] = 1/[(x+3)(x-1)]

Combine the terms:

[x^3 + 3X + 2x^2 - 3x + Xx + x^2 - 1] / [(x-1)(x+3)] = 1/[(x+3)(x-1)]

Step 8: Combine like terms and simplify Combine the terms in the numerator:

[x^3 + 3x^2 + 4X - 4x - 1] / [(x-1)(x+3)] = 1/[(x+3)(x-1)]

Now the equation becomes:

[x^3 + 3x^2 + 4X - 4x - 1] = 1

Step 9: Solve for x To solve for x, you would need to manipulate the equation further and find its roots. However, the equation at this stage is a cubic equation, which may not have a simple algebraic solution. You might need to use numerical methods or a calculator to approximate the roots.

Please note that solving cubic equations can be complex, and sometimes they do not have exact solutions in terms of elementary functions. If you're looking for numerical solutions, you can use methods like the Newton-Raphson method or software that can solve equations numerically.