%5E2-formula-proof.jpeg "(a-b)^2 Formula Proof")
(a-b)^2 Formula Proof
The formula (a-b)^2 = a^2 - 2ab + b^2 is a fundamental identity in algebra, and it has numerous applications in various branches of mathematics. In this article, we will provide a step-by-step proof of this formula.
The Proof
To prove the formula, we will start by expanding the left-hand side of the equation using the distributive property of multiplication over subtraction:
(a-b)^2 = (a-b)(a-b)
Now, we will multiply the two binomials:
(a-b)(a-b) = a(a-b) - b(a-b)
= a^2 - ab - ab + b^2
= a^2 - 2ab + b^2
Thus, we have shown that (a-b)^2 = a^2 - 2ab + b^2.
Alternative Proof
Another way to prove the formula is to use the FOIL method, which is a technique for multiplying two binomials. The FOIL method stands for "First, Outer, Inner, Last," which refers to the order in which we multiply the terms.
Using the FOIL method, we get:
(a-b)(a-b) = a(a) - a(b) - b(a) + b(b)
= a^2 - ab - ab + b^2
= a^2 - 2ab + b^2
Therefore, we have again shown that (a-b)^2 = a^2 - 2ab + b^2.
Conclusion
In conclusion, we have provided two different proofs of the formula (a-b)^2 = a^2 - 2ab + b^2. This formula is a powerful tool in algebra and has many applications in mathematics, physics, engineering, and other fields.
