Proving a^2 - b^2 = (a+b)(a-b) with Factorization

In summary, a^2 - b^2 = (a+b)(a-b) is proven by adding and subtracting ab to the expression a^2 - b^2, which results in (a+b)(a-b).
  • #1
scientifico
181
0
Hello, how can I prove using factorization that a^2 - b^2 = (a+b)(a-b) ?

a*a - b*b I should get something like a(a-b)+b(a-b) but how ?

thanks
 
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  • #2
a2 - b2 == a2 - b2 + ab - ab == a2 + ab - b2 - ab == a(a+b) - b(a+b) == ?
 
  • #3
Ok but how do I get a^2 - b^2 + ab - ab from a^2 - b^2 ?
 
  • #4
Add ab and subtract ab.
 
  • #5
scientifico said:
Ok but how do I get a^2 - b^2 + ab - ab from a^2 - b^2 ?

HallsofIvy said:
Add ab and subtract ab.

You can always add zero to any expression (here in the form of ab + (-ab), and you can always multiply an expression by 1.
 

FAQ: Proving a^2 - b^2 = (a+b)(a-b) with Factorization

1. What is factorization?

Factorization is the process of breaking down a mathematical expression into smaller, simpler parts. It involves finding the factors, or numbers that can be multiplied together to create the original expression.

2. Why is it important to prove a^2 - b^2 = (a+b)(a-b) with factorization?

Proving this identity is important because it helps to solidify our understanding of factorization and its applications in algebraic manipulations. It also serves as a foundation for more advanced concepts in mathematics.

3. How can I prove a^2 - b^2 = (a+b)(a-b) with factorization?

The proof can be done using the distributive property of multiplication and the fact that a^2 - b^2 can be rewritten as (a+b)(a-b). This can be shown by expanding (a+b)(a-b) and simplifying the resulting expression to get a^2 - b^2.

4. Can you provide an example of proving a^2 - b^2 = (a+b)(a-b) with factorization?

Sure, let's take the expression 9x^2 - 4y^2. By using the distributive property, we can rewrite this as (3x)^2 - (2y)^2. Now, using the identity (a+b)(a-b) = a^2 - b^2, we get (3x+2y)(3x-2y), which is equivalent to the original expression 9x^2 - 4y^2.

5. How is proving a^2 - b^2 = (a+b)(a-b) with factorization used in real life?

This identity is used in various applications, such as in simplifying algebraic expressions and solving equations. It is also used in physics and engineering to solve problems involving quadratic equations and in finance to calculate interest rates.

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