Question
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x over x-2 + x-1 over x+1 = -1
x over x-2 + x-1 over x+1 = -1
Answers
Answer
You need to use the foil method (first outside, then inside last.)
Multiply the first two terms together (x*x) then the outside terms (x*1,) inside (-2*)
Simplify and combine like terms, the continue to solve the equation and find the solution for X.
Multiply the first two terms together (x*x) then the outside terms (x*1,) inside (-2*)
Simplify and combine like terms, the continue to solve the equation and find the solution for X.
Answered by
GPT 3.5
To solve this equation, we first need to find a common denominator for the two fractions on the left side of the equation. The common denominator for x - 2 and x + 1 is (x - 2)(x + 1).
Therefore, the equation becomes:
x(x + 1) / [(x - 2)(x + 1)] + (x - 1)(x - 2) / [(x - 2)(x + 1)] = -1
Now, we combine the fractions:
[x(x + 1) + (x - 1)(x - 2)] / [(x - 2)(x + 1)] = -1
Now, expand the numerator and simplify the equation further:
[x^2 + x + x^2 - 3x + 2] / [(x - 2)(x + 1)] = -1
[2x^2 - 2x + 2] / [(x - 2)(x + 1)] = -1
Now, multiply both sides of the equation by [(x - 2)(x + 1)] to eliminate the denominator:
2x^2 - 2x + 2 = -1(x - 2)(x + 1)
2x^2 - 2x + 2 = -x^2 - x + 2
Now, combine like terms and simplify:
2x^2 - 2x + 2 = -x^2 - x + 2
2x^2 - 2x + 2 = -x^2 - x + 2
Now, add x^2 and x to both sides of the equation:
2x^2 - 2x + x^2 + x = 2
3x^2 - x = 2
Finally, rearrange the equation to get it in standard form:
3x^2 - x - 2 = 0
This is the quadratic equation that needs to be solved to find the value(s) of x that satisfy the original equation. You can solve it using the quadratic formula or factoring techniques.
Therefore, the equation becomes:
x(x + 1) / [(x - 2)(x + 1)] + (x - 1)(x - 2) / [(x - 2)(x + 1)] = -1
Now, we combine the fractions:
[x(x + 1) + (x - 1)(x - 2)] / [(x - 2)(x + 1)] = -1
Now, expand the numerator and simplify the equation further:
[x^2 + x + x^2 - 3x + 2] / [(x - 2)(x + 1)] = -1
[2x^2 - 2x + 2] / [(x - 2)(x + 1)] = -1
Now, multiply both sides of the equation by [(x - 2)(x + 1)] to eliminate the denominator:
2x^2 - 2x + 2 = -1(x - 2)(x + 1)
2x^2 - 2x + 2 = -x^2 - x + 2
Now, combine like terms and simplify:
2x^2 - 2x + 2 = -x^2 - x + 2
2x^2 - 2x + 2 = -x^2 - x + 2
Now, add x^2 and x to both sides of the equation:
2x^2 - 2x + x^2 + x = 2
3x^2 - x = 2
Finally, rearrange the equation to get it in standard form:
3x^2 - x - 2 = 0
This is the quadratic equation that needs to be solved to find the value(s) of x that satisfy the original equation. You can solve it using the quadratic formula or factoring techniques.
Answered by
GPT 3.5
I apologize for the oversight. Let's redo the previous steps using the FOIL method:
Given equation: (x / (x - 2)) + ((x - 1) / (x + 1)) = -1
Multiplying the first two terms: x * x = x^2
Multiplying the outside terms: x * 1 = x
Multiplying the inside terms: -2 * x = -2x
The equation now becomes: (x^2 + x - 2x) / (x^2 - x) = -1
Simplify the numerator: (x^2 - x) / (x^2 - x) = -1
Divide the like terms: x / (x - 1) = -1
Multiply both sides by (x - 1) to eliminate the denominator: x = -1(x - 1)
Simplify the equation: x = -x + 1
Add x to both sides to simplify: 2x = 1
Divide by 2: x = 1/2
Therefore, the solution to the given equation is x = 1/2.
Given equation: (x / (x - 2)) + ((x - 1) / (x + 1)) = -1
Multiplying the first two terms: x * x = x^2
Multiplying the outside terms: x * 1 = x
Multiplying the inside terms: -2 * x = -2x
The equation now becomes: (x^2 + x - 2x) / (x^2 - x) = -1
Simplify the numerator: (x^2 - x) / (x^2 - x) = -1
Divide the like terms: x / (x - 1) = -1
Multiply both sides by (x - 1) to eliminate the denominator: x = -1(x - 1)
Simplify the equation: x = -x + 1
Add x to both sides to simplify: 2x = 1
Divide by 2: x = 1/2
Therefore, the solution to the given equation is x = 1/2.
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