Question

If no denominator equals zero, which expression is equivalent to (x²+10x+25)/x+5 - (x²-6)/x-5

A. (2x²-19)/x-5
B. (2x²-19)/x²-25
C. 19/x-5
D. (-19)/x-5

Answer 1

To find an expression equivalent to (x²+10x+25)/x+5 - (x²-6)/x-5, we combine the fractions over a common denominator and simplify, leading to the final simplified form of (2x²-19)/x²-25, which is option B.

Step-by-step explanation:

The student's question involves simplifying a complex rational expression in algebra. Firstly, we need to combine the two fractions by finding a common denominator.

The first fraction, (x²+10x+25)/(x+5), can be recognized as a perfect square trinomial because it is the expansion of (x+5)², so its numerator simplifies to (x+5)(x+5).

The second fraction, (x²-6)/(x-5), does not have the same denominator.

To combine these, we multiply each fraction by a form of 1 that will give both fractions the same denominator, which in this case would be (x+5)(x-5) or x² - 25.

To avoid any potential mistakes, let's perform the operations step by step:

1. Expand the numerator of the first expression: (x+5)(x+5) = x² + 10x + 25.
2. Multiply the second expression's numerator and denominator by (x+5): ((x²-6)(x+5))/((x-5)(x+5)).
3. Combine both expressions over the common denominator (x-5)(x+5) or x² - 25.
4. Simplify the combined numerator by distributing and then subtracting the numerators.
5. Cancel out any like terms if possible.

Using these steps, we find that the simplified expression is:

²² + 10x + 25 - (x²-6)(x+5)

-----------------------

x² - 25

After simplifying the numerator and combining like terms, the final answer is (2x²-19)/x²-25, which corresponds to option B.