Question

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

Answer 1

Answer:

$\dfrac{12}{x-7},$

$x\neq 7,\ x\neq -3.$

Step-by-step explanation:

Consider the expression

$\dfrac{12x+36}{x^2-4x-21}.$

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

$x_{1,2}=\dfrac{-(-4)\pm \sqrt{100}}{2\cdot 1}=\dfrac{4\pm 10}{2}=7,\ -3.$

Then

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

Note that restrictions on the variable x are $x\neq 7,\ x\neq -3.$

3. Simplify the fraction:

$\dfrac{12x+36}{x^2-4x-21}=\dfrac{12(x+3)}{(x-7)(x+3)}=\dfrac{12}{x-7}.$