Question

Simplify completely 12x+36/x^2-4x-21 and find the restrictions on the variable

Answer 1

`(12)/(x-7),`

`x\\eq 7,\ x\\eq -3.`

Explanation:

Consider the expression

`(12x+36)/(x^2-4x-21).`

1. The numerator can be factored as

`12x+36=12(x+3).`

2. The denominator
`x^2-4x-21` has the discriminant

`D=(-4)^2-4\cdot (-21)\cdot 1=16+84=100.`

Then

`x_(1,2)=(-(-4)\pm √(100))/(2\cdot 1)=(4\pm 10)/(2)=7,\ -3.`

Then

`x^2-4x-21=(x-7)(x+3).`

Note that restrictions on the variable x are
`x\\eq 7,\ x\\eq -3.`

3. Simplify the fraction:

`(12x+36)/(x^2-4x-21)=(12(x+3))/((x-7)(x+3))=(12)/(x-7).`