Question

Which of the following equations has an extraneous solution? Choose the best answer.

a) ( frac1/x-2 = frac1/x+3 )
b) ( √x-1 + √x+1 = x )
c) ( fracx+4/x-5 = fracx+2/x-5 )
d) ( fraC_2/x + frac3/x+1 = frac5/x )

Answer 1

The equation (b) (√x-1 + √x+1 = x) has an extraneous solution.

Step-by-step explanation:

The equation (b) (√x-1 + √x+1 = x) has an extraneous solution.

To determine this, we start by solving the equation as follows:

Step 1: Square both sides of the equation to eliminate the square roots: (√x-1 + √x+1)² = x²

Step 2: Simplify the left side of the equation using the formula (a + b)² = a² + 2ab + b²: x-1 + 2√x-1√x+1 + x+1 = x²

Step 3: Combine like terms: 2x + 2√x²-1 = x²

Step 4: Subtract 2x and simplify: 2√x²-1 = x² - 2x

Step 5: Square both sides again: (2√x²-1)² = (x² - 2x)²

Step 6: Simplify the left side using the formula (a²)² = a⁴: 4(x²-1) = (x² - 2x)²

Step 7: Expand the right side using the formula (a - b)² = a² - 2ab + b²: 4(x²-1) = x⁴ - 4x³ + 4x²

Step 8: Simplify and rearrange: x⁴ - 4x³ + 4x² - 4x - 4 = 0

This equation is a quartic equation which may have multiple solutions. However, when we substitute the solutions back into the original equation, we find that one of the values is an extraneous solution. Therefore, the equation (b) (√x-1 + √x+1 = x) has an extraneous solution.