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### Solved Examples

**Question 1:**Find the sum of $\frac{x + 2}{x + 3}$ and $\frac{x + 2}{x - 2}$

**Solution:**

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

= $\frac{x^2 - 4 + x^2 + 3x + 2x + 6}{(x + 3)(x - 2)}$

= $\frac{2x^2 + 5x + 2}{(x + 3)(x - 2)}$

=> $\frac{x + 2}{x + 3}$ + $\frac{x + 2}{x - 2}$ = $\frac{2x^2 + 5x + 2}{(x + 3)(x - 2)}$.

= $\frac{x^2 - 4 + x^2 + 3x + 2x + 6}{(x + 3)(x - 2)}$

= $\frac{2x^2 + 5x + 2}{(x + 3)(x - 2)}$

=> $\frac{x + 2}{x + 3}$ + $\frac{x + 2}{x - 2}$ = $\frac{2x^2 + 5x + 2}{(x + 3)(x - 2)}$.

**Question 2:**Write a rational expression whose numerator is a quadratic polynomial with zeros -5 and $\frac{2}{4}$ and whose denominator is a quadratic polynomial with zeros $\frac{1}{3}$ and -2.

**Solution:**

Numerator of rational expression = (x + 5)(x - $\frac{2}{4}$ )

= (x + 5)($\frac{4x - 2}{4}$)

= $\frac{1}{4}$ (x + 5)(4x - 2)

= $\frac{1}{4}$ (4x

= $\frac{1}{4}$(4x

=> Numerator of rational expression = $\frac{1}{4}$(4x

Denominator of rational expression = (x + 2)(x - $\frac{1}{3}$)

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

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

= $\frac{1}{3}$ (3x

= $\frac{1}{3}$ (3x

=> Denominator of rational expression = $\frac{1}{3}$ (3x

=> Rational expression = $\frac{3(4x^2 + 18x - 10)}{4 (3x^2 + 5x - 2)}$.

= (x + 5)($\frac{4x - 2}{4}$)

= $\frac{1}{4}$ (x + 5)(4x - 2)

= $\frac{1}{4}$ (4x

^{2}- 2x + 20x - 10)= $\frac{1}{4}$(4x

^{2}+ 18x - 10)=> Numerator of rational expression = $\frac{1}{4}$(4x

^{2}+ 18x - 10)Denominator of rational expression = (x + 2)(x - $\frac{1}{3}$)

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

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

= $\frac{1}{3}$ (3x

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

^{2}+ 5x - 2)=> Denominator of rational expression = $\frac{1}{3}$ (3x

^{2}+ 5x - 2)=> Rational expression = $\frac{3(4x^2 + 18x - 10)}{4 (3x^2 + 5x - 2)}$.

**Question 3:**Express in lowest form, $\frac{2x^2 - x - 10}{2x^2 - 15x + 22}$.

**Solution:**

**Step 1:**

Factorized the polynomials

2x

^{2}- x - 10 = 2x

^{2}- 5x + 4x - 10

= x(2x - 5) + 2(2x - 5)

= (x + 2)(2x - 5)

=> 2x

^{2}- x - 10 = (x + 2)(2x - 5)

2x

^{2}- 15x + 22 = 2x

^{2}- 11x - 4x + 22

= x(2x - 11) - 2(2x - 11)

= (x - 2)(2x - 11)

=> 2x

^{2}- 15x + 22 = (x - 2)(2x - 11)

Step 2:

Step 2:

=> $\frac{2x^2 - x - 10}{2x^2 - 15x + 22}$ = $\frac{(x + 2)(2x - 5)}{(x - 2)(2x - 11)}$.