Question

Using quadratic equation ​

Answer 1

Explanation:

`(x+6)/(x+7)-(x+1)/(x+2)=((x+6)*(x+2))/((x+7)*(x+2))-((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

`(x+6)/(x+7)-(x+1)/(x+2)=((x+6)*(x+2))/((x+7)*(x+2))-((x+1)(x+7))/((x+2)(x+7))\\\\\\ =((x+6)*(x+2) - [(x+1)(x+7)])/((x+2)(x+7))\\\\\\= (x^(2)+8x+12 -[x^(2)+8x+7])/(x^(2)+9x+14)\\\\\\=(x^(2)+8x+12-x^(2)-8x-7)/(x^(2)+9x+14)\\\\=(5)/(x^(2)+9x+14)\\\\\\\\(x+6)/(x+7)-(x+1)/(x+2)=(1)/(3x+1)\\\\(5)/(x^(2)+9x+14)=(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 =
`√(2*2*5*i*i)=2i√(5)`

`x = (-b+√(D))/(2a) \ or (-b-√(D))/(2a)\\\\\\x= (-4+2i√(5))/(2) \ or (-4-2i√(5))/(2)\\\\x = (2*(-2+i√(5)))/(2) \ or (2*(-2-i√(5))/(2)\\\\x = -2 +i√(5) \ or -2-i√(5)`