in 9th grade work and writing Solve the following equation. Show all your work.
x/x−2 + x−1/x+1 = −1
To solve the equation x/x-2 + x-1/x+1 = -1, we first need to find a common denominator for the fractions on the left side of the equation.
The common denominator of x and x-2 is (x)(x-2) = x^2 - 2x.
The common denominator of x-1 and x+1 is (x-1)(x+1) = x^2 - 1.
So, the equation becomes:
x(x+1)/(x^2 - 2x) + (x-1)/(x^2 - 1) = -1
Now, we need to combine the fractions:
[x(x+1) + (x-1)(x-2)] / [(x^2 - 2x)(x^2 - 1)] = -1
Expanding the numerators:
[(x^2 + x) + (x^2 - 3x + 2)] / [(x^2 - 2x)(x^2 - 1)] = -1
(2x^2 - 2x + 2) / [(x^2 - 2x)(x^2 - 1)] = -1
Multiplying both sides by the denominators:
2x^2 - 2x + 2 = -[(x^2 - 2x)(x^2 - 1)]
2x^2 - 2x + 2 = -[x^4 - x^2 - 2x^2 + 2x]
2x^2 - 2x + 2 = -[-x^4 - 3x^2 + 2x]
2x^2 - 2x + 2 = x^4 + 3x^2 - 2x
Rearrange the equation:
x^4 + x^2 - 2 = 0
Now, this is a quadratic equation in terms of x^2. Let y = x^2:
y^2 + y - 2 = 0
(y + 2)(y - 1) = 0
Solving for y:
y1 = -2
y2 = 1
Substitute back in for y = x^2:
x^2 = -2
x^2 = 1
Taking the square root of both sides:
x = ±√(-2)
x = ±√(1)
So, the solution to the equation is x = √1 = 1 or x = -√1 = -1.
are you sure
Apologies for the mistake in the solution. Let's correct that.
Given equation: x/(x-2) + (x-1)/(x+1) = -1
First, find a common denominator for the fractions:
(x(x+1))/((x-2)(x+1)) + (x-1)(x-2)/(x-2)(x+1) = -1
Combine the fractions:
(x(x+1) + (x-1)(x-2))/((x-2)(x+1)) = -1
(x^2 + x + x^2 - 3x + 2)/((x-2)(x+1)) = -1
(2x^2 - 2x + 2)/((x-2)(x+1)) = -1
Multiply both sides by the common denominator:
2x^2 - 2x + 2 = -((x-2)(x+1))
Expand the right side:
2x^2 - 2x + 2 = -(x^2 - x - 2)
2x^2 - 2x + 2 = -x^2 + x + 2
Rearrange the equation:
3x^2 - 3x = 0
Factor out 3x:
3x(x-1) = 0
Set each factor to zero:
3x = 0 or x - 1 = 0
x = 0 or x = 1
Therefore, the solution to the equation is x = 0 or x = 1.
are you a 100% correct
Let's double-check the solution to ensure its accuracy.
Given equation: x/(x-2) + (x-1)/(x+1) = -1
First, find a common denominator for the fractions:
(x(x+1))/((x-2)(x+1)) + (x-1)(x-2)/(x-2)(x+1) = -1
Combine the fractions:
(x(x+1) + (x-1)(x-2))/((x-2)(x+1)) = -1
(x^2 + x + x^2 - 3x + 2)/((x-2)(x+1)) = -1
(2x^2 - 2x + 2)/((x-2)(x+1)) = -1
Multiply both sides by the common denominator:
2x^2 - 2x + 2 = -((x-2)(x+1))
Expand the right side:
2x^2 - 2x + 2 = -(x^2 - x - 2)
2x^2 - 2x + 2 = -x^2 + x + 2
Rearrange the equation:
3x^2 - 3x = 0
Factor out 3x:
3x(x-1) = 0
Set each factor to zero:
3x = 0 or x - 1 = 0
x = 0 or x = 1
Therefore, the solution to the equation is x = 0 or x = 1. The solution appears to be 100% correct.