Which of these is the extraneous solution for the equation given below? 2xx2−1−2x=x+1x2−1 A. x=−2 B x=−1 C. x=0 D. x=1

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Maaghu

Answered question

2021-11-17

Which of these is the extraneous solution for the equation given below?

\frac{2 x}{x^{2} - 1} - \frac{2}{x} = \frac{x + 1}{x^{2} - 1}

x = - 2

x = - 1

x = 0

x = 1

Answer & Explanation

Clis1955

Beginner2021-11-18Added 9 answers

Step 1: To Find
We have to find the extraneous solution for the equation

\frac{2 x}{x^{2} - 1} - \frac{2}{x} = \frac{x + 1}{x^{2} - 1}

Step 2: Calculation
Since we have

\frac{2 x}{x^{2} - 1} - \frac{2}{x} = \frac{x + 1}{x^{2} - 1}

\frac{2 x^{2} - 2 (x^{2} - 1)}{(x^{2} - 1) x} = \frac{x + 1}{x^{2} - 1}

\frac{2 x^{2} - 2 x^{2} + 2}{x} = \frac{x + 1}{1}

\frac{2}{x} = x + 1

2 = x (x + 1)

2 = x^{2} + x

x^{2} + x - 2 = 0

x^{2} + (2 - 1) x - 2 = 0

x^{2} + 2 x - x - 2 = 0

x (x + 2) - 1 (x + 2) = 0

(x + 2) (x - 1) = 0

x = - 2, 1

But 1 is excluded from the domain of the original equation because it would make the denominator of one of the fractions 0 and division by 0 is not allowed. Hence

x = 1

is an extraneous solution.
Therefore option (d) is correct.

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