Question

Solve and check:  3 x2 5x 6 x − 1 x 2 = 7 x 3

Answer 1

x = 7 is a solution.

x = -2 is an extraneous solution .

Answer 2

Here, the given expression,

`(3)/(x^2+5x+6)+(x-1)/(x+2)=(7)/(x+3)`,

`(3)/(x^2+3x+2x+6)+(x-1)/(x+2)=(7)/(x+3)`

`(3)/(x(x+3)+2(x+3))+(x-1)/(x+2)=(7)/(x+3)`

`(3)/((x+3)(x+2))+(x-1)/(x+2)=(7)/(x+3)`

`(3+(x-1)(x+3))/((x+3)(x+2)) = (7)/(x+3)`

`3+(x-1)(x+3)=7(x+2)`

`3+x^2-x+3x-3=7x+14`

`x^2+2x=7x+14`

`x^2+2x-7x-14=0`

`x^2-5x-14=0`

`x^2-7x+2x-14=0`

`x(x-7)+2(x-7)=0`

`(x+2)(x-7)=0`

`\implies x = -2\text{ or } x = 7`

For x = -2,

The right side of the given equation is not defined,

⇒ x = -2 can not be the solution of the given equation.

While for x = 7,

`(3)/(7^2+5* 7+6)+(7-1)/(7+2)=(7)/(7+3)`,

Hence, the solution of the given equation is,

x = 7.