Answer:
Option (D) is correct.
8 open parentheses x plus 4 close parentheses over quantity x minus 6
Step-by-step explanation:
Given : 8 x minus 56 over quantity x squared minus 49 divided by quantity x minus 6 over quantity x squared plus 11 x plus 28
We need to simplify above and choose one correct option out of given options.
First writing each term mathematically,
8 x minus 56 over quantity x squared minus 49 is written as , \(\frac{8x-56}{x^2-49}\)
quantity x minus 6 over quantity x squared plus 11 x plus 28 is written as,
\(\frac{x-6}{x^2+11x+28}\)
Combining, we get, 8 x minus 56 over quantity x squared minus 49 divided by quantity x minus 6 over quantity x squared plus 11 x plus 28 as
\(\frac{8x-56}{x^2-49}\div \frac{x-6}{x^2+11x+28}\)
Solving fraction separately, we get,
Consider first expression,
Applying identity \(a^2-b^2=(a+b)(a-b)\)
\(\Rightarrow \frac{8x-56}{x^2-49}=\frac{8x-56}{(x+7)(x-7)}\)
taking 8 common from numerator,
\(\Rightarrow \frac{8x-56}{(x+7)(x-7)}=\frac{8(x-7)}{(x+7)(x-7)}\)
On simplifying , we get,
\(\Rightarrow \frac{8(x-7)}{(x+7)(x-7)}=\frac{8}{(x+7)}\)
Consider second expression, \(\frac{x-6}{x^2+11x+28}\)
Solving quadratic using middle term splitting method,
\(\Rightarrow \frac{x-6}{x^2+11x+28}=\frac{x-6}{x^2+7x+4x+28}\)
\(\Rightarrow \frac{x-6}{x^2+7x+4x+28}= \frac{x-6}{(x+4)(x+7)}\)
Combining,
\(\frac{8x-56}{x^2-49}\div \frac{x-6}{x^2+11x+28}\)
\(\Rightarrow \frac{8}{(x+7)}\div \frac{x-6}{(x+4)(x+7)}\)
\(\Rightarrow \frac{8}{(x+7)}\times \frac{(x+4)(x+7)}{x-6}\)
On solving,
\(\Rightarrow 8 \times \frac{(x+4)}{x-6}\)
Thus, we obtain (D) is as correct option.
8 open parentheses x plus 4 close parentheses over quantity x minus 6
