Question

Simplify completely quantity 12 x plus 36 over quantity x squared minus 4 x minus 21 and find the restrictions on the variable.

a.) 12 over quantity x minus 7, x ≠ 7
b.) 12 over quantity x minus 7, x ≠ 7, x ≠ −3
c.) quantity x plus 3 over quantity x minus 7, x ≠ 7
d.)quantity x plus 3 over quantity x minus 7, x ≠ 7, x ≠ −3

Answer 1

Option B

12 over quantity x minus 7, x ≠ 7, x ≠ −3

Solution:

Given that we have to simplify quantity 12 x plus 36 over quantity x squared minus 4 x minus 21

So we have to simplify,

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

We have to find the restrictions on the variable

The above given Rational function is defined, for

${x}^{2} - 4x - 21\ne0\\\\(x+3)(x-7)\ne0\\\\x\ne -3,x\ne7$

In order to simplify the above expression, we need to factor both the numerator and the denominator

$\rightarrow \frac{12x + 36}{ {x}^{2} - 4x - 21 } = \frac{12(x + 3)}{ {x}^{2} - 4x - 21}$

For the denominator, we need to split the middle term of the quadratic to get factored form,

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

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

Fcatoring the denominator part,

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

Cancel out common factors to get,

$\frac{12x + 36}{ {x}^{2} - 4x - 21 } = \frac{12}{x - 7} \\\\where\\\\x\ne -3,x\ne7$

Thus option B is correct. 12 over quantity x minus 7, x ≠ 7, x ≠ −3