Question 108·Hard·Equivalent Expressions
Which of the following expressions is equivalent to
for all real for which the expression is defined?
For rational-expression equivalence questions, start by factoring denominators to see their structure and domain restrictions. Choose a common denominator (usually the product of distinct factors), rewrite each fraction with that denominator, and then combine them by adding or subtracting the numerators, being very careful to distribute any minus signs across entire parentheses. After simplifying the numerator, you can factor out a negative sign and rewrite the denominator (for example, turning into ) to match the form of an answer choice, always ensuring you have not canceled factors that would change where the expression is undefined.
Hints
Look at the denominators
Notice that is a difference of squares. How can you factor ? What does that tell you about a good common denominator for the two fractions?
Give both fractions the same denominator
Try to rewrite so that it has denominator . What can you multiply its numerator and denominator by to make that happen?
Be careful subtracting the numerators
Once both fractions have denominator , combine them into one fraction by subtracting the numerators. Remember the minus sign applies to the entire second numerator, not just the first term.
Adjust signs to match an answer choice
After simplifying the combined numerator, see if you can factor out from the fraction. How does changing into affect the sign of the fraction?
Desmos Guide
Enter the original expression
In Desmos, type a function like f(x) = (2x^2 - 5x - 3)/(x^2 - 4) - (3x + 1)/(x + 2) to graph the original expression. Note there will be vertical asymptotes (or breaks) where the expression is undefined.
Enter each answer choice as a separate function
On new lines, enter A(x) = (x^2 + 1)/(x^2 - 4), B(x) = (x^2 + 1)/(x^2 + 4), C(x) = (x^2 - 1)/(4 - x^2), and D(x) = (x^2 + 1)/(4 - x^2) (or just label them differently).
Compare graphs and domains
Zoom out and compare each candidate graph with . The correct choice will have a graph that lies exactly on top of everywhere they are both defined and will have breaks/asymptotes at the same -values as . Any option whose graph does not perfectly coincide with (or has different undefined points) is not equivalent.
Step-by-step Explanation
Factor the denominator and note the common denominator
Notice that the first denominator is a difference of squares:
This shows that and make the denominator zero, so the whole expression is undefined at those values. The natural common denominator for both fractions is .
Rewrite the second fraction with denominator
The second fraction has denominator . To give it denominator , multiply its numerator and denominator by :
Now the whole expression becomes
Combine the fractions and simplify the numerator
Since the denominators are the same, subtract the numerators:
First expand :
Now subtract:
So the expression simplifies to
Factor out the negative and match an answer choice
Factor from the numerator:
Now multiply the numerator and denominator by (which does not change the value of the expression):
This matches choice D, , which is therefore the correct equivalent expression for all real where the original expression is defined.