%5E2-formula.jpeg "(ab+bc+cd)^2 Formula")
The Formula for (ab+bc+cd)^2: A Detailed Explanation
In algebra, one of the most important formulas to learn is the expansion of (ab+bc+cd)^2. This formula is commonly used in various mathematical problems, particularly in algebra and calculus. In this article, we will explore the formula, its proof, and some examples to illustrate its application.
The Formula
The formula for (ab+bc+cd)^2 is given by:
(ab+bc+cd)^2 = a^2b^2 + 2ab^2c + 2abc^2 + b^2c^2 + 2bc^2d + 2bcd^2 + c^2d^2
Proof
To prove this formula, we can start by expanding the square of the binomial expression (ab+bc+cd) using the distributive property of multiplication over addition.
(ab+bc+cd)^2 = (ab+bc+cd)(ab+bc+cd)
Expanding the right-hand side of the equation, we get:
(ab+bc+cd)(ab+bc+cd) = a^2b^2 + ab^2c + abc^2 + abbc + abc^2 + bc^2d + bcd^2 + cd^2
Combining like terms, we arrive at the final formula:
a^2b^2 + 2ab^2c + 2abc^2 + b^2c^2 + 2bc^2d + 2bcd^2 + c^2d^2
Examples
Example 1
Expand (2x+3y+4z)^2.
Using the formula, we get:
(2x+3y+4z)^2 = (2x)^2(3y)^2 + 2(2x)^2(3y) + 2(2x)(3y)^2 + (3y)^2(4z)^2 + 2(3y)^2(4z) + 2(3y)(4z)^2 + (4z)^2
Simplifying the expression, we get:
(2x+3y+4z)^2 = 4x^2y^2 + 12x^2y + 12xy^2 + 9y^2z^2 + 24y^2z + 24yz^2 + 16z^2
Example 2
Expand (a+b+c)^2.
Using the formula, we get:
(a+b+c)^2 = a^2b^2 + 2ab^2c + 2abc^2 + b^2c^2 + 2bc^2c + 2bcc^2 + c^2c^2
Simplifying the expression, we get:
(a+b+c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2
Conclusion
In conclusion, the formula for (ab+bc+cd)^2 is a powerful tool in algebra and calculus. By understanding and applying this formula, students and mathematicians can solve a wide range of problems and simplify complex expressions.
