How do you simplify #n/(2n+10)+1/(n^2-25)#?

Question

Eva Adamson · Accepted Answer

#(n^2-5n+2)/(2(n-5)(n+5))# #"before we can add the fractions we require them to have"# #"a "color(blue)"common denominator"# #"first factorise the denominators"# #rArrn/(2(n+5))+1/((n-5)(n+5))larrcolor(blue)" difference of squares"# #"multiply the numerator/denominator of the fraction on the "# #" left by " (n-5)# #"multiply the numerator/denominator of the fraction on the"# #"right by " 2# #rArr(n(n-5))/(2(n+5)(n-5))+(2)/(2(n+5)(n-5))# #"now the denominators are common we can add the numerators"# #"leaving the denominator as it is."# #rArr(n^2-5n+2)/(2(n+5)(n-5))# #=(n^2-5n+2)/(2(n+5)(n-5))#

Eli Assouline · Answer

undefined To simplify the expression n/(2n+10) + 1/(n^2-25), we can follow these steps:

Step 1: Factor the denominators.
The denominator 2n+10 can be factored as 2(n+5), and the denominator n^2-25 can be factored as (n+5)(n-5).

Step 2: Find the least common denominator (LCD).
The LCD is the product of the distinct factors from both denominators, which in this case is 2(n+5)(n-5).

Step 3: Rewrite the fractions with the LCD as the denominator.
n/(2n+10) can be rewritten as n/2(n+5), and 1/(n^2-25) can be rewritten as 1/(n+5)(n-5).

Step 4: Multiply each fraction by the appropriate factor to eliminate the denominators.
For n/2(n+5), we multiply the numerator and denominator by 2(n-5), and for 1/(n+5)(n-5), we multiply the numerator and denominator by 2.

Step 5: Simplify the resulting expression.
After performing the multiplication, the expression becomes (n(n-5) + 2) / 2(n+5)(n-5).

Therefore, the simplified expression is (n^2 - 5n + 2) / 2(n+5)(n-5).