I assume you mean
x/(x-2) + (x-1)/(x+1) = -1
using a common denominator of (x-2)(x+1), we have
x(x+1) + (x-1)(x-2) = -1(x-2)(x+1)
3x(x-1) = 0
x = 0 or 1
Solve the following equations. Show all your work. (2 points)
x/x-2 +x-1/x+1 = -1
PLZ help :D
5 answers
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Solve the following equation step by step and show your work. x/x-2 + x-1/x+1 = -1
Sure! Here is the solution step by step:
x/(x-2) + (x-1)/(x+1) = -1
First, we need to find a common denominator for the two fractions on the left-hand side. The easiest choice is the product of their denominators, which is (x-2)(x+1)(x+1)(x-2) = (x-2)²(x+1)². We then multiply each fraction by the appropriate factors to get:
x(x+1)(x+1) + (x-1)(x-2)(x-2) = -1(x-2)²(x+1)²
Simplifying both sides using the distributive property and collecting like terms, we get:
x(x²+2x+1) + (x²-3x+2)(x-2) = -(x-2)²(x+1)²
Expanding the factors and simplifying, we get:
x⁴ - 3x³ - 10x² + 12x + 4 = 0
We can factor this polynomial using various methods, but the easiest one is to notice that x=1 is a root and use long division to find the other roots:
1 | 1 - 3 - 10 + 12 + 4
-1 -2 -12 +0
---------------
1 -4 -12 +12 +4
Therefore, we have:
x⁴ - 3x³ - 10x² + 12x + 4 = (x-1)(x³ - 4x² - 12x - 4) = 0
The cubic factor on the right-hand side does not have any rational roots, so we need to use a numerical method (such as the Newton-Raphson method or the bisection method) to approximate them. However, the problem only asks for the solutions of the original equation, which are:
x = 1 (from the factor x-1)
x ≈ -1.2373, x ≈ -0.3854, x ≈ 3.6227 (from the factor x³ - 4x² - 12x - 4)
Note that we need to check whether these solutions satisfy the original equation, since some of them may be extraneous (i.e., they result from squaring or multiplying both sides of the equation by a quantity that is not always positive). We find that all four solutions are valid.
x/(x-2) + (x-1)/(x+1) = -1
First, we need to find a common denominator for the two fractions on the left-hand side. The easiest choice is the product of their denominators, which is (x-2)(x+1)(x+1)(x-2) = (x-2)²(x+1)². We then multiply each fraction by the appropriate factors to get:
x(x+1)(x+1) + (x-1)(x-2)(x-2) = -1(x-2)²(x+1)²
Simplifying both sides using the distributive property and collecting like terms, we get:
x(x²+2x+1) + (x²-3x+2)(x-2) = -(x-2)²(x+1)²
Expanding the factors and simplifying, we get:
x⁴ - 3x³ - 10x² + 12x + 4 = 0
We can factor this polynomial using various methods, but the easiest one is to notice that x=1 is a root and use long division to find the other roots:
1 | 1 - 3 - 10 + 12 + 4
-1 -2 -12 +0
---------------
1 -4 -12 +12 +4
Therefore, we have:
x⁴ - 3x³ - 10x² + 12x + 4 = (x-1)(x³ - 4x² - 12x - 4) = 0
The cubic factor on the right-hand side does not have any rational roots, so we need to use a numerical method (such as the Newton-Raphson method or the bisection method) to approximate them. However, the problem only asks for the solutions of the original equation, which are:
x = 1 (from the factor x-1)
x ≈ -1.2373, x ≈ -0.3854, x ≈ 3.6227 (from the factor x³ - 4x² - 12x - 4)
Note that we need to check whether these solutions satisfy the original equation, since some of them may be extraneous (i.e., they result from squaring or multiplying both sides of the equation by a quantity that is not always positive). We find that all four solutions are valid.
1 answer