Give step-by-step solution with explanation and final answer:
{given the answer without using bullet-

Question

VIPSolutions · Accepted Answer

# Problem

Solve for $ x $:

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

# Step-by-Step Solution

First, find a common denominator. The least common denominator is $ (x+2)(x-2) $.

Rewrite each term with the common denominator:

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

Combine the numerators:

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

Expand the numerators:

$$
4(x-2) + 3(x+2) = 4x - 8 + 3x + 6 = (4x + 3x) + (-8 + 6) = 7x - 2
$$

So the equation is:

$$
\frac{7x - 2}{(x+2)(x-2)} = 1
$$

Multiply both sides by $ (x+2)(x-2) $:

$$
7x - 2 = (x+2)(x-2)
$$

Expand the right side:

$$
(x+2)(x-2) = x^2 - 4
$$

So:

$$
7x - 2 = x^2 - 4
$$

Rearrange terms to one side:

$$
 = x^2 - 4 - 7x + 2
$$
$$
 = x^2 - 7x - 2
$$

So,

$$
x^2 - 7x - 2 = 
$$

This is a quadratic equation. Use the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
where $ a=1 $, $ b=-7 $, $ c=-2 $:

$$
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-2)}}{2(1)}
$$
$$
x = \frac{7 \pm \sqrt{49 + 8}}{2}
$$
$$
x = \frac{7 \pm \sqrt{57}}{2}
$$

# Final Answer

$$
\boxed{
x = \frac{7 + \sqrt{57}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{57}}{2}
}
$$

Both values are valid as they do not make the denominators zero ($ x \neq 2, -2 $).

✨ VIPSolutions