X + 1/(14 - X) = 1/(8X)

Steps to Solve

1. Multiply through by common denominators

To eliminate the fractions, multiply both sides by the least common denominator, which is $8X(14 - X)$.

$$ (8X(14 - X)) \left( X + \frac{1}{14 - X} \right) = (8X(14 - X)) \left( \frac{1}{8X} \right) $$

This simplifies to:

$$ 8X^2(14 - X) + 8 = 14 - X $$

2. Distribute and simplify the equation

Now, distribute on the left side:

$$ 8X^2(14) - 8X^3 + 8 = 14 - X $$

Rearranging terms gives us:

$$ -8X^3 + 8X^2 \cdot 14 + X - 14 + 8 = 0 $$

3. Combine like terms

Combine like terms to simplify the polynomial:

$$ -8X^3 + 112X^2 + X - 6 = 0 $$

4. Use synthetic division or numerical methods to factor the cubic

We can either use synthetic division, numerical methods, or graphing to solve.

Assuming one solution is $X = 1$, we divide:

$$ -8(1)^3 + 112(1)^2 + (1) - 6 = 0 $$

5. Finding remaining roots

Having found one root, we can now use polynomial division or factorization to find other roots.

The equation can be factored as:

$$ -8(X - 1)(X^2 + 14.25X + 0.75) = 0 $$

6. Solve for remaining roots using the quadratic formula

Now, apply the quadratic formula $X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 14.25$, and $c = 0.75$.

This will yield the other solutions for $X$.