Solve Rational Equation: 1/(x+1)² + 1/(x+1) = 1

Solve the following equation:

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

To solve this problem, we'll follow these steps:

• Step 1: Clear fractions by multiplying through by the least common denominator.
• Step 2: Simplify the resulting equation.
• Step 3: Solve the quadratic equation using the quadratic formula.

Now, let's work through each step:
**Step 1:** Multiply both sides by $(x+1)^2$ to clear the denominators:
$(x+1)^2 \left(\frac{1}{(x+1)^2} + \frac{1}{x+1}\right) = (x+1)^2 \cdot 1$
This simplifies to:
$1 + (x+1) = (x+1)^2$
**Step 2:** Simplify the equation:
$1 + x + 1 = x^2 + 2x + 1$
Combine like terms:
$2 + x = x^2 + 2x + 1$
Rearrange to form a quadratic equation:
$x^2 + 2x + 1 - x - 2 = 0$
Thus, we have:
$x^2 + x - 1 = 0$
**Step 3:** Solve the quadratic equation $x^2 + x - 1 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 1$, and $c = -1$.
Calculate the discriminant:
$b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-1) = 1 + 4 = 5$
Thus, $x$ is:
$x = \frac{-1 \pm \sqrt{5}}{2}$
**Conclusion:** The solutions to the equation are:
$x = \frac{-1 + \sqrt{5}}{2}$ and $x = \frac{-1 - \sqrt{5}}{2}$
Upon verifying with given choices, the correct answer is:
$x = -\frac{1}{2}[1\pm\sqrt{5}\rbrack$