To solve the equation:

$\frac{4}{x} - \frac{5}{x+3} = 1$

Step 1: Eliminate the denominators

Multiply through by the least common denominator (LCD), which is $x(x+3)$, to get rid of the fractions:

$x(x+3) \cdot \left( \frac{4}{x} - \frac{5}{x+3} \right) = x(x+3) \cdot 1$

Simplifying each term:

$4(x+3) - 5x = x(x+3)$

Step 2: Expand both sides

Expand both sides of the equation:

$4x + 12 - 5x = x^2 + 3x$

Step 3: Simplify the equation

Combine like terms on the left side:

$-x + 12 = x^2 + 3x$

Step 4: Move all terms to one side

Rearrange the equation to form a quadratic equation:

$x^2 + 3x + x - 12 = 0$

This simplifies to:

$x^2 + 4x - 12 = 0$

Step 5: Solve the quadratic equation

Use the quadratic formula to solve for $x$, where $a = 1$, $b = 4$, and $c = -12$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}$

$x = \frac{-4 \pm \sqrt{16 + 48}}{2}$

$x = \frac{-4 \pm \sqrt{64}}{2}$

$x = \frac{-4 \pm 8}{2}$

This gives two solutions:

$x = \frac{-4 + 8}{2} = 2 \quad \text{or} \quad x = \frac{-4 - 8}{2} = -6$

Step 6: Check for extraneous solutions

Substitute both values of $x$ back into the original equation to ensure they are valid solutions.

• For $x = 2$:
  $\frac{4}{2} - \frac{5}{2+3} = 1$ $2 - 1 = 1 \quad \text{(Valid solution)}$

• For $x = -6$:
  $\frac{4}{-6} - \frac{5}{-6+3} = 1$ This results in division by zero, so $x = -6$ is not a valid solution.

Final Answer:

The solution is:

$x = 2$

Do you want more details or have any questions?

Here are 5 related questions:

1. How do you solve quadratic equations using factorization?
2. What is the quadratic formula and how is it derived?
3. How can you check if a quadratic equation has real or complex solutions?
4. What is an extraneous solution and why does it occur?
5. How do you solve rational equations without the quadratic formula?

Tip: Always check your solutions in the original equation, especially for rational equations, to avoid extraneous solutions.