Question

show your solution
B. Perform the indicated operation.
$\frac{2 \times - 1}{ {2x}^{2} + 5x + 3 } + \frac{2}{3x + 3}$​

Answer 1

$\frac{10x + 3}{3 (x + 1)(2x + 3)}$

Question :

$\frac{2x - 1}{2 {x}^{2} + 5x + 3 } + \frac{2}{3x + 3}$

Solution :

$= \frac{2x - 1}{2 {x}^{2} + 5x + 3} + \frac{2}{3x + 3}$

Let's find the roots of quadratic equation.

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

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

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

Cross Multiplying the terms

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

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

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

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

$= \frac{10 {x}^{2} + 13x + 3 }{(2x + 3)(x + 1)(3x + 3)}$

Let's find the roots of quadratic equation,

$= \frac{10 {x}^{2} + 10x + 3x + 3 }{(2x + 3)(x + 1)(3x + 1)}$

$= \frac{10x(x + 1) + 3(x + 1)}{(2x + 3)(x + 1)(3x + 1)}$

$= \frac{(10x + 3)(x + 1)}{(2x + 3)(x + 1)(3x + 1)}$

$= \frac{(10x + 3)}{(2x + 3)(3x + 3)}$

$= \frac{10x + 3}{3 (x + 1)(2x + 3)}$

Answer 2

$\frac{(10x+3)}{3(x+1)(2x+3)}$

Given :

$\frac{2x-1}{2x^{2} +5x+3} +\frac{2}{3x+3}$

Solution :

$\implies \frac{2x-1}{2x^{2} +5x + 3} +\frac{2}{3x+3}$

Find roots of the quadratic equation ,

$\implies \frac{2x-1}{2x^{2} + 2x + 3x +5 } + \frac{2}{3x+3}$

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

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

Cross multiply the equations,

$\implies \frac{(2x1)(3x+3)+2(2x+3)(x+1)}{(2x+3)(x+1)(3x+3)}$

$\implies \frac{6x^{2}+6x-3x-3+4x^{2} +4x+6x+6 }{(2x+3)(x+1)(3x+3)}$

$\implies \frac{10x^{2} +13x+3}{(2x+3)(x+1)(3x+3)}$

$\implies \frac{10x^{2} +10x+3x+3}{(2x+3)(x+1)(3x+3)}$

$\implies \frac{10x(x+1)+3(x+1)}{(2x+3)(3x+3)(x+1)}$

$\implies \frac{(10x+3)(x+1)}{(2x+3)(3x+3)(x+1)}$

$\implies \frac{(10x+3)}{3(x+1) (2x+3)}$