%5E2-formula.jpeg "(ab+bc-2ac)^2 Formula")
The Formula: (ab+bc-2ac)^2
In algebra, we often come across expressions that involve variables and constants. One such expression is (ab+bc-2ac)^2, which is a quadratic expression that can be simplified and solved using various algebraic techniques. In this article, we will explore the formula, its expansion, and some examples to illustrate its application.
Expansion of the Formula
The formula (ab+bc-2ac)^2 can be expanded using the binomial theorem, which states that:
(a+b)^2 = a^2 + 2ab + b^2
In this case, we have:
(ab+bc-2ac)^2 = (ab)^2 + 2(ab)(bc) + (bc)^2 - 4(ab)(2ac) - 4(bc)(2ac) + (2ac)^2
Simplifying the expression, we get:
(ab+bc-2ac)^2 = a^2b^2 + 2a^2bc - 4a^2bc + b^2c^2 + 4ab^2c - 8abc^2 + 4a^2c^2
Simplification of the Formula
The expanded formula can be simplified further by combining like terms:
(ab+bc-2ac)^2 = a^2b^2 - 2a^2bc + b^2c^2 + 4ab^2c - 8abc^2 + 4a^2c^2
Examples
Example 1
If a = 2, b = 3, and c = 4, find the value of (ab+bc-2ac)^2.
Substituting the values, we get:
(2*3+3*4-2*2*4)^2 = (6+12-16)^2 = 2^2
(ab+bc-2ac)^2 = 4
Example 2
If a = 1, b = 2, and c = 3, find the value of (ab+bc-2ac)^2.
Substituting the values, we get:
(1*2+2*3-2*1*3)^2 = (2+6-6)^2 = 2^2
(ab+bc-2ac)^2 = 4
Conclusion
In conclusion, the formula (ab+bc-2ac)^2 is a quadratic expression that can be expanded and simplified using algebraic techniques. By combining like terms, we can simplify the expression to obtain a more compact form. The formula has various applications in mathematics, physics, and engineering, and its understanding is essential for solving complex problems.
