Which expression is equivalent to ?
When a quadratic expression is already given and the answer choices are factored forms, first look for special patterns like the difference of squares: check if both terms are perfect squares and the middle sign is a minus. Rewrite each term as a square, apply the formula , and then match your factored form to the correct answer choice. You can quickly eliminate wrong options by checking whether their expanded form would give the correct constant term and whether they introduce an unwanted middle (linear) term.
Hints
Look for a common factoring pattern
Ask yourself if looks like a special pattern you know, such as a square, a difference of squares, or something with a common factor.
Think about square numbers
Notice that and are both perfect squares. What are their square roots, and how can you rewrite the expression using those roots?
Use a known formula
If an expression looks like , recall the factoring formula that works for this pattern and apply it to .
Desmos Guide
Enter the original expression
In Desmos, type y = 81v^2 - 225 (use x instead of v if needed, like y = 81x^2 - 225). This is your reference graph.
Compare each option to the original
For each answer choice, enter its expression as another function, for example y = (9x - 15)*(9x + 15) for one option. The equivalent expression will produce a graph that lies exactly on top of the graph of y = 81x^2 - 225 for all visible x-values.
Step-by-step Explanation
Recognize the pattern
Look at and notice it has the form "something squared minus something squared." This is called a difference of squares, which factors using a special formula.
Rewrite each term as a square
Rewrite each part:
- because .
- because .
So the expression can be seen as .
Apply the difference of squares formula
Use the formula for a difference of squares:
Here, and , so
Among the choices, this matches .