Question

Using quadratic equation ​

Answer 1

Answer:

Step-by-step explanation:

$\frac{x+6}{x+7}-\frac{x+1}{x+2}=\frac{(x+6)*(x+2)}{(x+7)*(x+2)}-\frac{(x+1)(x+7)}{(x+2)(x+7)}$

USE FOIL method

(x +6)(x + 2) = x*x + x*2 +6*x +2*6

                  =x² + 2x + 6x + 12    {Combine like terms}

                  = x² + 8x + 12

(x+ 7)(x + 2) = x*x+ x*2 +7*x + 7*2

                  = x² +2x+ 7x + 14

                 = x² + 9x + 14

(x+1)*(x +7) = x*x + x *7 + 1*x + 1*7

                =x² + 7x +x + 7

                =x² + 8x + 7

$\frac{x+6}{x+7}-\frac{x+1}{x+2}=\frac{(x+6)*(x+2)}{(x+7)*(x+2)}-\frac{(x+1)(x+7)}{(x+2)(x+7)}\\\\\\ =\frac{(x+6)*(x+2) - [(x+1)(x+7)]}{(x+2)(x+7)}\\\\\\= \frac{x^{2}+8x+12 -[x^{2}+8x+7]}{x^{2}+9x+14}\\\\\\=\frac{x^{2}+8x+12-x^{2}-8x-7}{x^{2}+9x+14}\\\\=\frac{5}{x^{2}+9x+14}\\\\\\\\\frac{x+6}{x+7}-\frac{x+1}{x+2}=\frac{1}{3x+1}\\\\\frac{5}{x^{2}+9x+14}=\frac{1}{3x+1}$

Cross multiply,

5*(3x+1) = x² + 9x + 14

5*3x + 5*1 = x² + 9x + 14

15x + 5 = x² + 9x + 14

x² + 9x + 14 - 5x - 5 = 0

x² + 9x - 5x + 14 - 5 = 0

Combine like terms

x² + 4x  + 9 = 0

a = 1 ; b = 4 ; c = 9

D = b² - 4ac

  = 4² - 4*1*9

  = 16 - 36

  = -20

  = 20i²    {i² = -1}

√D = √20i² = $\sqrt{2*2*5*i*i}=2i\sqrt{5}$

$x = \frac{-b+\sqrt{D}}{2a} \ or \frac{-b-\sqrt{D}}{2a}\\\\\\x= \frac{-4+2i\sqrt{5}}{2} \ or \frac{-4-2i\sqrt{5}}{2}\\\\x = \frac{2*(-2+i\sqrt{5})}{2} \ or \frac{2*(-2-i\sqrt{5}}{2}\\\\x = -2 +i\sqrt{5} \ or -2-i\sqrt{5}$