Question

Simplify; 1/x^2+5x+6 + 1/x^2+3x+2 pls I need a step by step solution urgently

Answer 1

Answer:

The simplest form of  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$  is  $\frac{2}{(x+1)(x+3)}$

Step-by-step explanation:

To add two algebraic fractions do these steps

1. Factorize each denominator
2. Simplify each fraction to its lowest term
3. Find the LCM of the two denominators
4. Divide LCM by each denominator and multiply the numerators by its corresponding quotients
5. Add the two fraction and simplify the last answer

To simplify $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$

Factorize each denominator

∵ The denominator of the first fraction is x² + 5x + 6

∵ x² = (x)(x)

∵ 6 = (2)(3)

∵ (2)(x) + (3)(x) = 5x ⇒ middle term

∴ x² + 5x + 6 = (x + 2)(x + 3)

∵ The denominator of the second fraction is x² + 3x + 2

∵ x² = (x)(x)

∵ 2 = (2)(1)

∵ (2)(x) + (1)(x) = 3x ⇒ middle term

∴ x² + 3x + 2 = (x + 2)(x + 1)

Find the LCM of the two denominators

∵ The denominators are (x + 2)(x + 3) and (x + 2)(x + 1)

- LCM is all the different factors multiplied together

∴ LCM of them is (x + 1)(x + 2)(x + 3)

- Divide LCM by each denominator

∵ (x + 1)(x + 2)(x + 3) ÷ (x + 2)(x + 3) = (x + 1)

- Multiply the numerator of the first fraction by (x + 1)

∵ (x + 1)(x + 2)(x + 3) ÷ (x + 2)(x + 1) = (x + 3)

- Multiply the numerator of the second fraction by (x + 3)

∴  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$ = $\frac{x+1}{(x+1)(x+2)(x+3)}+\frac{x+3}{(x+1)(x+2)(x+3)}$

- Add the numerators and write the answer as a single fraction

∴  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$ = $\frac{2x+4}{(x+1)(x+2)(x+3)}$

- Factorize the numerator by taking 2 as a common factor

∵ 2x + 4 = 2(x + 2)

∴  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$ = $\frac{2(x+2)}{(x+1)(x+2)(x+3)}$

- Simplify the fraction by dividing up and down by (x + 2)

∴  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$ = $\frac{2}{(x+1)(x+3)}$

The simplest form of  $\frac{1}{x^{2}+5x+6}+\frac{1}{x^{2}+3x+2}$  is  $\frac{2}{(x+1)(x+3)}$