Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. Solve the following equation and. Show all your work x/x-2+x-1/x+1=-1
x/x-2+x-1/x+1=-1
To solve this equation, we need to simplify the expression on the left side by finding a common denominator.
The common denominator is (x-2)(x+1):
x(x+1)/(x-2)(x+1) + (x-1)(x-2)/(x-2)(x+1) = -1
Now we can combine the fractions:
x(x+1) + (x-1)(x-2) = -1(x-2)(x+1)
Simplifying the left side:
x^2 + x + x^2 - 3x + 2 = -x^2 + x + 2x - 2
Simplifying the right side:
-x^2 - x + 2x + 2 = -x^2 + x - 2
Bringing all the terms to one side:
2x^2 - 2x - 4 = 0
Dividing both sides by 2:
x^2 - x - 2 = 0
This is a quadratic equation, which we can factor:
(x-2)(x+1) = 0
Therefore, the solutions are x = 2 and x = -1.
However, we need to check if these solutions are valid, because we might have introduced extraneous solutions when we found the common denominator.
Checking x = 2:
2/0 + 1/3 = -1
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The left side is undefined when x = 2, so this solution is extraneous and should be discarded.
Checking x = -1:
-1/-3 + 0/0 = -1
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The left side is still undefined when x = -1, so this solution is also extraneous and should be discarded.
Therefore, the equation has no valid solutions.
I will assume you meant:
x/(x-2) + (x-1)/(x+1) = -1
multiply each term by (x-2)(x+1)
x(x+1) + (x-1)(x-2) = -(x-2)(x+1)
x^2 + x + x^2 - 3x + 2 = -x^2 + x + 2
3x^2 - 3x = 0
3x(x - 1) = 0
x = 0 or x = 1
I think the bot messed up the expansion of the right side of
x(x+1) + (x-1)(x-2) = -(x-2)(x+1)
I apologize for my mistake in the expansion of the right side of the equation. Thank you for providing the correct expansion and solution. You are correct that the solutions are x = 0 and x = 1, and we should check them for validity, which can be done by plugging in the values into the original equation.
To solve the equation (x / (x - 2)) + ((x - 1) / (x + 1)) = -1, we'll follow these steps:
Step 1: Simplify the fractions by finding a common denominator.
=> Common denominator = (x - 2)(x + 1)
=> (x / (x - 2)) + ((x - 1) / (x + 1)) = (-1 * (x - 2)(x + 1)) / (x - 2)(x + 1)
Step 2: Combine the fractions on the left side.
=> (x(x + 1) + (x - 1)(x - 2)) / (x - 2)(x + 1) = (-1 * (x - 2)(x + 1)) / (x - 2)(x + 1)
Step 3: Expand and simplify the numerator on both sides.
=> [x^2 + x + (x^2 - 3x + 2)] / (x - 2)(x + 1) = (-x^2 + x - 2) / (x - 2)(x + 1)
Step 4: Combine like terms in the numerator.
=> (2x^2 - 2x + 2) / (x - 2)(x + 1) = (-x^2 + x - 2) / (x - 2)(x + 1)
Step 5: Cross-multiply to eliminate the denominators.
=> (2x^2 - 2x + 2) * (x - 2)(x + 1) = (-x^2 + x - 2) * (x - 2)(x + 1)
Step 6: Expand and simplify both sides of the equation.
=> (2x^2 - 2x + 2) * [(x - 2)(x + 1)] = (-x^2 + x - 2) * [(x - 2)(x + 1)]
=> 2x^2 - 2x + 2 = -x^4 + 5x^3 - 2x^2 - 12x + 4
Step 7: Rearrange the equation to form a quadratic equation.
=> 0 = -x^4 + 5x^3 - 4x^2 - 10x + 2
Step 8: Solve the quadratic equation using factoring, graphing, or numerical methods.
Unfortunately, the given equation is a quartic equation (4th degree polynomial), and its analytical solutions are quite complex. Therefore, it is impractical to solve this equation using step-by-step methods. We recommend using a graphing calculator or numerical approximation methods (such as Newton's method or bisection method) to find an approximate solution for this equation.
To solve the equation:
x/x-2 + x-1/x+1 = -1
First, let's find the least common denominator (LCD) for the fractions. The LCD is the product of the denominators, which is:
LCD = (x-2)(x+1)
Next, we will multiply every term in the equation by the LCD to eliminate the fractions. This step is called clearing the denominators. Multiplying both sides of the equation by the LCD, we get:
[(x-2)(x+1)] * (x/x-2) + [(x-2)(x+1)] * (x-1/x+1) = -1 * [(x-2)(x+1)]
Simplifying and distributing, we have:
(x+1) * x + (x-1) * (x-2) = -1 * (x-2)(x+1)
Expanding these expressions:
x^2 + x + x^2 - 3x + 2 = -x^2 + 3x - 2
Combining like terms:
2x^2 - 3x + 2 = -x^2 + 3x - 2
Moving all terms to one side of the equation, we get:
2x^2 - 3x + x^2 - 3x + 2 = 0
Combining like terms again:
3x^2 - 6x + 2 = 0
Now, we can solve this quadratic equation. To factor or use the quadratic formula, we need to calculate the discriminant first. The discriminant is given by:
Discriminant (D) = b^2 - 4ac
In this case, a = 3, b = -6, and c = 2. Substituting these values into the formula, we have:
D = (-6)^2 - 4 * 3 * 2 = 36 - 24 = 12
Since the discriminant is positive (D > 0), the quadratic equation has two real solutions.
From here, we can solve the quadratic equation using either factoring or the quadratic formula. Let's use the quadratic formula, which states that for an equation in the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± sqrt(D)) / (2a)
Substituting the values from our equation, we have:
x = (-(-6) ± sqrt(12)) / (2 * 3)
Simplifying further:
x = (6 ± sqrt(12)) / 6
Now, we can simplify the square root:
x = (6 ± 2 * sqrt(3)) / 6
Next, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2 in this case:
x = (3 ± sqrt(3)) / 3
Therefore, the solutions to the equation are:
x = (3 + sqrt(3)) / 3 and x = (3 - sqrt(3)) / 3
These are the final answers to the equation.