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Questions and Answers
What is the result of factoring the expression $a^2 - b^2$?
What is the result of factoring the expression $a^2 - b^2$?
- $a - b$
- ($a + b)($a - b)$ (correct)
- $a^2 + b^2$
- $a + b$
Which of the following is NOT a consequence of applying the difference of squares formula?
Which of the following is NOT a consequence of applying the difference of squares formula?
- It allows simplification of quadratic equations.
- It results in a perfect square trinomial. (correct)
- It can be used to solve for real roots.
- It is applicable to complex numbers.
When can the difference of squares formula be applied?
When can the difference of squares formula be applied?
- Only in polynomial equations.
- Only when both terms are positive.
- When one term is a square and the other is also a square. (correct)
- When both terms are negative.
If the expression $x^2 - 16$ is factored, what are the factors?
If the expression $x^2 - 16$ is factored, what are the factors?
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Which expression represents the incorrect application of the difference of squares concept?
Which expression represents the incorrect application of the difference of squares concept?
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Study Notes
Difference of Squares Formula
- The expression ( a^2 - b^2 ) factors into ( (a - b)(a + b) ).
- This formula is applicable when both terms are perfect squares and are subtracted.
Consequences of the Difference of Squares
- Valid outcomes include recognizing that ( x^2 - y^2 = (x - y)(x + y) ).
- It is NOT a consequence that ( a^2 - b^2 ) can be wrongfully associated with other algebraic identities like ( (a + b)^2 ).
Application Conditions
- The difference of squares formula can be applied when one term is a perfect square and another is also a perfect square, specifically in the format ( a^2 - b^2 ).
- It is essential that the operation is subtraction for this factorization to hold.
Example of Factoring
- The expression ( x^2 - 16 ) can be factored as ( (x - 4)(x + 4) ).
- Here, ( 16 ) is recognized as ( 4^2 ), confirming its nature as a perfect square.
Incorrect Application
- Misapplication might involve factoring something like ( a^2 + b^2 ) using the difference of squares, which is invalid as it does not follow the required subtraction form.
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