perform the indicated operations and simplify x-5/x-7-x-1/x+7+x-35/x^2-49
If what you typed means
(x-5)/(x-7) - (x-1)/(x+7) + (x-35)/(x^2-49) then
put everything over a common denominator. In this case, since x^2 - 49 = (x-7)(x+7) that will be our LCD.
The numerator will then be:
(x-5)(x+7) - (x-1)(x-7) + (x-35)
= x^2 + 2x - 35 - (x^2 - 8x + 7) + (x - 35)
= (11x - 77)
= 11(x-7)
Put that over (x-7)(x+7) and you have
11(x-7) / (x-7)(x+7)
= 11/(x+7)
You need to clarify your formula by spacing and/or parentheses.
x - 5/x - 7 - x - 1/x + 7 + x - 35/x^2 - 49 (?)
x - 5/(x-7) - x - 1/(x-7) + x 35/(x^2-49) (?)
To perform the indicated operations and simplify the expression, let's break it down step by step.
Given expression: (x - 5)/(x - 7) - (x - 1)/(x + 7) - (x - 35)/(x^2 - 49)
Step 1: Simplify each individual fraction.
The first fraction, (x - 5)/(x - 7), is already in its simplest form.
To simplify the second fraction, (x - 1)/(x + 7), we need to find the least common denominator (LCD). In this case, it is (x + 7). So, we multiply the numerator and denominator of the second fraction by (x + 7) to get:
(x - 1)/(x + 7) * (x + 7)/(x + 7) = (x - 1)(x + 7)/(x + 7)
For the third fraction, (x - 35)/(x^2 - 49), let's factor the denominator, x^2 - 49. It can be written as the difference of squares: (x - 7)(x + 7). Now we have:
(x - 35)/((x - 7)(x + 7))
Step 2: Find the LCD for the three fractions.
The LCD is the product of the distinct factors and the highest powers of each factor. In this case, the LCD is (x - 7)(x + 7).
Step 3: Rewrite each fraction with the LCD as the denominator.
For the first fraction, (x - 5)/(x - 7), the LCD is already the denominator, so it remains the same.
For the second fraction, we need to multiply both the numerator and denominator by (x - 7) to get:
((x - 1)(x + 7))/(x + 7)(x - 7)
The third fraction already has the denominator in the desired form.
Step 4: Combine the fractions.
Now that all the fractions have the same denominator, we can subtract them. The expression becomes:
[(x - 5) - (x - 1)(x + 7)]/(x + 7)(x - 7) - (x - 35)((x - 7)(x + 7))/(x + 7)(x - 7)
Step 5: Simplify the expression.
Let's simplify the numerator first:
(x - 5) - (x - 1)(x + 7) = x - 5 - (x^2 + 6x - 7) = x - 5 - x^2 - 6x + 7 = -x^2 - 5x + 2
Next, simplify the denominator:
(x + 7)(x - 7) = x^2 - 49
Putting it all together, the simplified expression is:
(-x^2 - 5x + 2)/(x^2 - 49)
And there you have it! The expression (x - 5)/(x - 7) - (x - 1)/(x + 7) - (x - 35)/(x^2 - 49) simplifies to (-x^2 - 5x + 2)/(x^2 - 49).