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 , [tex]\frac{8x-56}{x^2-49}[/tex]
quantity x minus 6 over quantity x squared plus 11 x plus 28 is written as,
[tex]\frac{x-6}{x^2+11x+28}[/tex]
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
[tex]\frac{8x-56}{x^2-49}\div \frac{x-6}{x^2+11x+28}[/tex]
Solving fraction separately, we get,
Consider first expression,
Applying identity [tex]a^2-b^2=(a+b)(a-b)[/tex]
[tex]\Rightarrow \frac{8x-56}{x^2-49}=\frac{8x-56}{(x+7)(x-7)}[/tex]
taking 8 common from numerator,
[tex]\Rightarrow \frac{8x-56}{(x+7)(x-7)}=\frac{8(x-7)}{(x+7)(x-7)}[/tex]
On simplifying , we get,
[tex]\Rightarrow \frac{8(x-7)}{(x+7)(x-7)}=\frac{8}{(x+7)}[/tex]
Consider second expression, [tex]\frac{x-6}{x^2+11x+28}[/tex]
Solving quadratic using middle term splitting method,
[tex]\Rightarrow \frac{x-6}{x^2+11x+28}=\frac{x-6}{x^2+7x+4x+28}[/tex]
[tex]\Rightarrow \frac{x-6}{x^2+7x+4x+28}= \frac{x-6}{(x+4)(x+7)}[/tex]
Combining,
[tex]\frac{8x-56}{x^2-49}\div \frac{x-6}{x^2+11x+28}[/tex]
[tex]\Rightarrow \frac{8}{(x+7)}\div \frac{x-6}{(x+4)(x+7)}[/tex]
[tex]\Rightarrow \frac{8}{(x+7)}\times \frac{(x+4)(x+7)}{x-6}[/tex]
On solving,
[tex]\Rightarrow 8 \times \frac{(x+4)}{x-6}[/tex]
Thus, we obtain (D) is as correct option.
8 open parentheses x plus 4 close parentheses over quantity x minus 6
