subtract the two rational expressions and write your answer in simplest form...\n\n$\\frac{x}{x + 4}…
subtract the two rational expressions and write your answer in simplest form...\n\n$\\frac{x}{x + 4} - \\frac{x + 36}{x^2 - 16}$
Answer
Explanation:
Step1: Factor the denominators
Factor the quadratic expression $x^2 - 16$ using the difference of squares. $$x^2 - 16 = (x - 4)(x + 4)$$
Step2: Find the common denominator
Identify the least common denominator (LCD) for the two expressions. $$\text{LCD} = (x + 4)(x - 4)$$
Step3: Rewrite expressions with LCD
Multiply the numerator and denominator of the first term by $(x - 4)$. $$\frac{x(x - 4)}{(x + 4)(x - 4)} - \frac{x + 36}{(x + 4)(x - 4)}$$
Step4: Combine the numerators
Subtract the second numerator from the first over the common denominator. $$\frac{x^2 - 4x - (x + 36)}{(x + 4)(x - 4)}$$
Step5: Simplify the numerator
Distribute the negative sign and combine like terms. $$\frac{x^2 - 5x - 36}{(x + 4)(x - 4)}$$
Step6: Factor the resulting numerator
Find two numbers that multiply to $-36$ and add to $-5$. $$\frac{(x - 9)(x + 4)}{(x + 4)(x - 4)}$$
Step7: Cancel common factors
Divide out the common factor $(x + 4)$ from the numerator and denominator. $$\frac{x - 9}{x - 4}$$
Answer:
$$\frac{x - 9}{x - 4}$$