Question

2x/x+3 + 1/x-1 = 4/x^2+2x-3 решите уравнение

Answer 1

2x/x+3 + 1/x-1 = 4/x^2+2x-3
x≠-3
x≠1
2x/x+3+1/x-1-4/x²+2x-2=0
2x/x=3+1/x-1-4/x²+3x-x-3=0
2x/x+3+1/x-1-4/x(x+)-(x+3)=0
2x/x+3+1/x-1-4/(x+3)(x-1)=0
2x²-2x+x+3-4/(x+3)(x-1)=0
2x²+x-2x-1/(x+3)(x-1)=0
x(2x+1)-(2x+1)/(x+3)(x-1)=0
(2x+1)(x-1)/(x+3)(x-1)=0
2x+1/x+3=0
2x+1=0
2x=-1
x=-1/2
x≠-3
x≠1
x=-1/2

Answer 2

To solve the equation (2x)/(x+3) + 1/(x-1) = 4/(x^2+2x-3), follow these steps:

Step 1: Factorize the denominators (x+3) and (x-1) and the quadratic expression (x^2 + 2x - 3).

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

So, the equation becomes:

(2x)/(x + 3) + 1/(x - 1) = 4/((x + 3)(x - 1))

Step 2: Find a common denominator for both fractions on the left side. The common denominator will be (x + 3)(x - 1).

Step 3: Rewrite the equation with the common denominator.

(2x(x - 1) + (x + 3))/(x + 3)(x - 1) = 4/(x + 3)(x - 1)

Step 4: Eliminate the denominator by multiplying both sides of the equation by (x + 3)(x - 1).

(2x(x - 1) + (x + 3)) = 4

Step 5: Expand and simplify the equation.

2x^2 - 2x + x + 3 = 4

Step 6: Combine like terms.

2x^2 - x + 3 = 4

Step 7: Move all terms to one side of the equation to make it equal to zero.

2x^2 - x + 3 - 4 = 0

2x^2 - x - 1 = 0

Step 8: Now, you can solve this quadratic equation. There are several ways to do it, but the most straightforward way is by factoring (if possible) or by using the quadratic formula.

However, the equation 2x^2 - x - 1 = 0 does not factor easily, so we will use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Where a = 2, b = -1, and c = -1.

x = (1 ± √((-1)^2 - 4 * 2 * -1)) / 2 * 2

x = (1 ± √(1 + 8)) / 4

x = (1 ± √9) / 4

x = (1 ± 3) / 4

Now, we have two possibilities:

1. x = (1 + 3) / 4 = 4 / 4 = 1
2. x = (1 - 3) / 4 = -2 / 4 = -1/2

So, the solutions to the equation are x = 1 and x = -1/2.