Solve the Rational Equation: (x³+1)/(x-1)² = x+4

Solve the following equation:

$\frac{x^3+1}{(x-1)^2}=x+4$

To solve this equation, we follow these steps:

• Step 1: Multiply both sides by $(x-1)^2$ to eliminate the fraction.
• Step 2: Expand and simplify both sides of the equation.
• Step 3: Rearrange the equation to form a polynomial equal to zero.
• Step 4: Solve the resulting polynomial using factorization or the quadratic formula.

Now, let's execute these steps:

Step 1: Multiply both sides by $(x-1)^2$:
$(x^3 + 1) = (x + 4)(x - 1)^2$

Step 2: Expand the right side:
$(x + 4)(x^2 - 2x + 1) = x(x^2 - 2x + 1) + 4(x^2 - 2x + 1)$

Calculating each part yields:
$x(x^2 - 2x + 1) = x^3 - 2x^2 + x$
$4(x^2 - 2x + 1) = 4x^2 - 8x + 4$

Add these together:
$x^3 - 2x^2 + x + 4x^2 - 8x + 4 = x^3 + 2x^2 - 7x + 4$

Step 3: Combine terms and rearrange:
$x^3 + 1 = x^3 + 2x^2 - 7x + 4$

Simplify by cancelling $x^3$ from both sides:
$1 = 2x^2 - 7x + 4$

Move 1 to the right side:
$0 = 2x^2 - 7x + 3$

Step 4: Solve the quadratic equation $2x^2 - 7x + 3 = 0$.

Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = -7$, and $c = 3$.

Calculate the discriminant:
$b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot 3 = 49 - 24 = 25$

Now plug into the quadratic formula:
$x = \frac{7 \pm \sqrt{25}}{4}$

Simplify:
$x = \frac{7 \pm 5}{4}$

Two solutions arise:
$x = \frac{12}{4} = 3$ and $x = \frac{2}{4} = \frac{1}{2}$

Since $x = 1$ would make the denominator zero, it is not a valid solution for the original equation.

Therefore, the solution to the problem is $x = 3$ or $x = \frac{1}{2}$.