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Difficulty: MediumRadical and Rational Equations
An equation is shown below.
x+1x−2+1x=6x2−2x\frac{x+1}{x-2} + \frac{1}{x} = \frac{6}{x^2-2x}
If xx satisfies the equation above, what is the value of xx?
  1. A
    -2
  2. B
    2
  3. -4Answer
  4. D
    4

Answer

-4
The correct answer is −4-4. To solve the rational equation, we first identify the least common denominator as x(x−2)x(x-2). Multiplying both sides by this expression eliminates the denominators, yielding x(x+1)+(x−2)=6x(x+1) + (x-2) = 6. Expanding and simplifying gives the quadratic equation x2+2x−8=0x^2 + 2x - 8 = 0, which factors as (x+4)(x−2)=0(x+4)(x-2) = 0. This yields potential solutions of x=−4x = -4 and x=2x = 2. However, substituting x=2x = 2 back into the original equation results in division by zero, making x=2x = 2 extraneous. The only valid solution is −4-4.

Step-by-Step Solution

1
Find the least common denominator (LCD) of the rational expressions in the equation.
The denominators are x−2x-2, xx, and x2−2xx^2-2x. Since x2−2x=x(x−2)x^2-2x = x(x-2), the LCD is x(x−2)x(x-2), with the restriction that x≠0x \neq 0 and x≠2x \neq 2.
Finding a common denominator allows us to clear the fractions by multiplying both sides of the equation.
2
Multiply each term of the equation by the LCD to eliminate the denominators.
x(x+1)+1(x−2)=6x(x+1) + 1(x-2) = 6
This clears the denominators and converts the rational equation into a polynomial equation.
3
Expand and simplify the resulting equation.
x2+x+x−2=6x^2 + x + x - 2 = 6, which simplifies to x2+2x−2=6x^2 + 2x - 2 = 6.
Combining like terms prepares the equation to be written in standard quadratic form.
4
Rearrange the terms into standard quadratic form: ax2+bx+c=0ax^2 + bx + c = 0.
x2+2x−8=0x^2 + 2x - 8 = 0
Subtracting 66 from both sides sets the quadratic equation to zero so it can be solved by factoring.
5
Factor the quadratic equation.
(x+4)(x−2)=0(x+4)(x-2) = 0
Finding two numbers that multiply to −8-8 and add to 22 (44 and −2-2) allows us to solve for xx.
6
Find the potential solutions by setting each factor to zero.
x=−4x = -4 or x=2x = 2
Applying the zero product property gives the potential roots of the quadratic equation.
7
Check the potential solutions in the original equation to identify any extraneous solutions.
For x=−4x = -4, the equation is defined and valid. For x=2x = 2, the denominators x−2x-2 and x2−2xx^2-2x become zero, which is undefined.
Rational equations can produce extraneous solutions that make the original denominators zero, so they must be checked and discarded.

Key Concept

Solving rational equations by finding a common denominator and checking for extraneous solutions.
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Topics

Radical and Rational Equations

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