Determine whether the given sequences are equal to each other. (k² - 9) and (k + 3)(k - 3)
Understand the Problem
The question is asking us to determine if the expression (k² - 9) is equal to the expression (k + 3)(k - 3). We will do this by simplifying both expressions and comparing the results.
Answer
The expressions are equal, \( k^2 - 9 = (k + 3)(k - 3) \).
Answer for screen readers
The expressions ( k^2 - 9 ) and ( (k + 3)(k - 3) ) are equal.
Steps to Solve
- Simplifying the first expression
We have the expression ( k^2 - 9 ). This is a difference of squares, which can be factored into:
$$ k^2 - 9 = (k - 3)(k + 3) $$
- Simplifying the second expression
Now we look at the expression ( (k + 3)(k - 3) ). This expression is already factored, so we can leave it as is for comparison.
- Comparing the two expressions
Now we compare our two results from steps 1 and 2. We have:
$$ k^2 - 9 = (k - 3)(k + 3) $$ and $$ (k + 3)(k - 3) $$
Both expressions simplify to the same factored form, confirming they are equal.
The expressions ( k^2 - 9 ) and ( (k + 3)(k - 3) ) are equal.
More Information
The expression ( k^2 - 9 ) is a classic example of the difference of squares. This method is commonly used in algebra to simplify equations and solve quadratic expressions.
Tips
- Forgetting to recognize that ( k^2 - 9 ) can be factored into ( (k - 3)(k + 3) ).
- Not simplifying both expressions before comparing them for equality.
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