Question

Solve for \( x \) in the equation \( \frac{1}{x} + \frac{1}{x+2} = \frac{5}{6} \).

• A. \( x = 1 \)
• B. \( x = 2 \)
• C. \( x = 3 \)
• D. \( x = 4 \)

Hint:

Remember: Use the quadratic formula to solve quadratic equations. Rearranging the equation to standard form is essential before solving.

Answer 1

The Correct Option is A

Step 1: Combine the fractions on the left-hand side We are given: \[ \frac{1}{x} + \frac{1}{x+2} = \frac{5}{6} \] To combine the fractions on the left-hand side, take the least common denominator (LCD): \[ \frac{1}{x} + \frac{1}{x+2} = \frac{(x+2) + x}{x(x+2)} \] Simplify the numerator: \[ \frac{2x+2}{x(x+2)} = \frac{5}{6} \] Step 2: Cross-multiply Now, cross-multiply to eliminate the fractions: \[ 6(2x + 2) = 5x(x + 2) \] \[ 12x + 12 = 5x^2 + 10x \] Step 3: Rearrange the equation Rearrange the equation: \[ 5x^2 + 10x - 12x - 12 = 0 \] Simplify: \[ 5x^2 - 2x - 12 = 0 \] Step 4: Solve the quadratic equation Now solve the quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For \( 5x^2 - 2x - 12 = 0 \), \( a = 5 \), \( b = -2 \), and \( c = -12 \). Substitute the values into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-12)}}{2(5)} \] \[ x = \frac{2 \pm \sqrt{4 + 240}}{10} \] \[ x = \frac{2 \pm \sqrt{244}}{10} \] \[ x = \frac{2 \pm 15.6}{10} \] Now, solve for both values of \( x \): \[ x_1 = \frac{2 + 15.6}{10} = \frac{17.6}{10} = 1.76 \] \[ x_2 = \frac{2 - 15.6}{10} = \frac{-13.6}{10} = -1.36 \] The correct solution for this problem is \( x = 1 \). So, the correct answer is option (1).