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Expanding the Square of a Binomial: (4 + 7a^2)^2
In mathematics, understanding how to expand expressions involving squares of binomials is crucial. This involves applying the distributive property or using a specific formula. Let's explore how to expand the expression (4 + 7a^2)^2.
The Formula for Squaring a Binomial
The formula for squaring a binomial (a + b) is:
(a + b)^2 = a^2 + 2ab + b^2
This formula tells us that to square a binomial, we need to square the first term, add twice the product of the first and second term, and finally add the square of the second term.
Applying the Formula to (4 + 7a^2)^2
Let's identify 'a' and 'b' in our expression:
- a = 4
- b = 7a^2
Now, we can apply the formula:
(4 + 7a^2)^2 = 4^2 + 2 * 4 * 7a^2 + (7a^2)^2
Simplifying the Expression
Let's simplify the expression step by step:
- 4^2 = 16
- 2 * 4 * 7a^2 = 56a^2
- (7a^2)^2 = 49a^4
Finally, combine the terms to get the expanded form:
(4 + 7a^2)^2 = 16 + 56a^2 + 49a^4
Conclusion
Therefore, the expanded form of (4 + 7a^2)^2 is 16 + 56a^2 + 49a^4. This process demonstrates how to apply the formula for squaring a binomial to simplify complex expressions.