Solve the Rational Expression: Find x in (x³+1)/(x+1)²=x

Solve the following equation:

$\frac{x^3+1}{(x+1)^2}=x$

To solve the equation $\frac{x^3 + 1}{(x+1)^2} = x$, we will follow these steps:

• Step 1: Set up the equation for solving by cross-multiplying.
• Step 2: Simplify and solve the resulting polynomial equation.
• Step 3: Solve for the values of $x$.

Let's work through the solution:

Step 1: Cross-multiply to eliminate the fraction:

Expand the right-hand side:

Step 2: Set the expanded equation equal:

Cancel $x^3$ from both sides:

Re-arrange the equation to form a standard quadratic equation:

Step 3: Solve the quadratic equation using the quadratic formula:

The quadratic formula is:

Substitute the values of $a$, $b$, and $c$ into the formula:

Calculate the discriminant and simplify:

Simplify further:

This gives the solutions:

Since $x = -1$ would make the denominator zero, it is not allowed as a solution. Thus, the only valid solution is:

Therefore, the solution to the equation is $x = \frac{1}{2}$.