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Which of the following is the correct expansion of the algebraic identity (a+b)2?
- a)a2+2ab+b2
- b)a2+b2
- c)a2−2ab+b2
- d)2a2+2b2
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
Which of the following is the correct expansion of the algebraic ident...
The algebraic identity (a+b)2 = a2+2ab+b2
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Which of the following is the correct expansion of the algebraic ident...
Understanding the Algebraic Identity
The expression (a + b)² is an important algebraic identity used to simplify and expand quadratics. Let's break down the correct expansion of this identity.
Expanding (a + b)²
To expand (a + b)², we can apply the formula:
- (a + b)² means (a + b) multiplied by itself: (a + b)(a + b).
Now, we use the distributive property (also known as the FOIL method for binomials):
- First: Multiply the first terms: a * a = a²
- Outer: Multiply the outer terms: a * b = ab
- Inner: Multiply the inner terms: b * a = ab
- Last: Multiply the last terms: b * b = b²
Now, we combine these results:
- a² + ab + ab + b² = a² + 2ab + b²
This confirms that the correct expansion is:
Correct Answer
- a² + 2ab + b² (Option A)
Why Other Options Are Incorrect
- Option B: a² + b² only represents the squares of a and b, missing the cross-product (2ab).
- Option C: a² - 2ab + b² is the expansion of (a - b)², not (a + b)².
- Option D: 2a² + 2b² adds coefficients that do not fit the identity.
Conclusion
The expansion of (a + b)² is a fundamental concept in algebra. Recognizing the correct terms and their coefficients is essential for solving more complex algebraic problems.
The expression (a + b)² is an important algebraic identity used to simplify and expand quadratics. Let's break down the correct expansion of this identity.
Expanding (a + b)²
To expand (a + b)², we can apply the formula:
- (a + b)² means (a + b) multiplied by itself: (a + b)(a + b).
Now, we use the distributive property (also known as the FOIL method for binomials):
- First: Multiply the first terms: a * a = a²
- Outer: Multiply the outer terms: a * b = ab
- Inner: Multiply the inner terms: b * a = ab
- Last: Multiply the last terms: b * b = b²
Now, we combine these results:
- a² + ab + ab + b² = a² + 2ab + b²
This confirms that the correct expansion is:
Correct Answer
- a² + 2ab + b² (Option A)
Why Other Options Are Incorrect
- Option B: a² + b² only represents the squares of a and b, missing the cross-product (2ab).
- Option C: a² - 2ab + b² is the expansion of (a - b)², not (a + b)².
- Option D: 2a² + 2b² adds coefficients that do not fit the identity.
Conclusion
The expansion of (a + b)² is a fundamental concept in algebra. Recognizing the correct terms and their coefficients is essential for solving more complex algebraic problems.
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Which of the following is the correct expansion of the algebraic ident...
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Which of the following is the correct expansion of the algebraic identity (a+b)2?a)a2+2ab+b2b)a2+b2c)a2−2ab+b2d)2a2+2b2Correct answer is option 'A'. Can you explain this answer?
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