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Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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What is the simplified value of (a+b)2−(a−b)2?
2ab
4ab
2a2+2b2
a2−b2
4ab
Substitute small values for variables (e.g., let a=2,b=1). Then (2+1)2−(2−1)2=32−12=9−1=8. Checking the options: 4(2)(1)=8. This matches Option B.
An algebraic expression consisting of the difference between the squares of two binomials: (a + b)2 and (a - b)2.
(a+b)2−(a−b)2=4ab
Substitute small values for variables (e.g., let a=2,b=1). Then (2+1)2−(2−1)2=32−12=9−1=8. Checking the options: 4(2)(1)=8. This matches Option B.
Many students fail to distribute the negative sign to the (−b)2 term, incorrectly expanding (a−b)2 as a2−b2 or forgetting to change the internal sign.
Expand the first square
Apply the identity (a+b)2=a2+2ab+b2 to the first term of the expression.
(a+b)2=a2+2ab+b2
Expand the second square
Apply the identity (a−b)2=a2−2ab+b2 to the second term.
(a−b)2=a2−2ab+b2
Subtract the expansions
Subtract the result of the second expansion from the first while carefully distributing the negative sign.
(a2+2ab+b2)−(a2−2ab+b2)=a2+2ab+b2−a2+2ab−b2
Simplify terms
Cancel out the identical terms a2 and b2 which have opposite signs.
2ab+2ab=4ab
B is correct because the expansion of (a+b)2−(a−b)2 consistently results in 4ab after canceling common terms.
This identity is a fundamental case of the Difference of Squares, which appears frequently in factoring quadratic expressions and simplifying complex rational functions.