Discovering Patterns in the Difference of Squares Equation

In summary, the conversation was about factoring equations and noticing patterns in the coefficients. The individual pointed out how certain terms compensated for each other and how the signs affected the overall result. They also discussed a method of grouping terms to make the pattern more visible.
  • #1
rocomath
1,755
1
I'm looking for patterns and if you can add to things I noticed before working it out, that would be good :-]

1. [tex](a+b+c)(a+b-c)=a^2+b^2+c^2+2ab[/tex]

I noticed that b+c and b-c compensated for each other.

2. [tex](a+b+c)(a-b-c)=a^2-b^2-c^2-2bc[/tex]

a+b and a-b compensated for each other and the fact that it's b+c and -b-c, is the reason that it was -2bc?

3. [tex](a+b-c)(a-b+c)=a^2-b^2-c^2+2bc[/tex]

a+b and a-b compensated for each other, Now I figured from problem 2 that it would be 2bc again, but I didn't predict the sign correctly?
 
Last edited:
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  • #2
Not sure I will be helpful here but all I can see is that
rocophysics said:
2. [tex](a+b+c)(a-b-c)=a^2-b^2-c^2-2bc[/tex]

a+b and a-b compensated for each other and the fact that it's b+c and -b-c, is the reason that it was -2bc?

[tex](a+b+c)(a-b-c)==(a+(b+c))(a-(b+c))=(a)^2-(b+c)^2[/tex]

and the same for the 3rd one.

for the first one:
[tex]
(a+b+c)(a+b-c)((a+b)+c)((a+b)-c)[/tex]

EDIT: oh wait...that is not what you were talking about...my bad
 
  • #3
rock.freak667 said:
EDIT: oh wait...that is not what you were talking about...my bad
Nope, lol. But I didn't even think about what you were doing (grouping then putting it in a more visible manner). Thanks, still helped!
 

1. What is the difference of squares?

The difference of squares is a mathematical expression that involves subtracting the square of one number from the square of another number. It can be written as (a^2 - b^2) and is a common algebraic expression used to simplify equations.

2. How do you factor a difference of squares?

To factor a difference of squares, you need to identify the two perfect squares being subtracted. Then, use the formula (a^2 - b^2) = (a + b)(a - b) to factor the expression. For example, if you have 4x^2 - 9, you can factor it as (2x + 3)(2x - 3).

3. What is the significance of the difference of squares?

The difference of squares is significant because it is a fundamental concept in algebra and is often used to simplify and solve equations. It also has applications in geometry and can be used to find the side lengths of a right triangle.

4. Can a difference of squares be negative?

Yes, a difference of squares can be negative. This occurs when one of the squares is negative, and the other is positive. For example, in the expression 9 - 16, the difference of squares is -7.

5. How is the difference of squares related to the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This can be represented as (a^2 + b^2 = c^2). The difference of squares formula is a special case of this theorem when one of the sides is equal to 0. For example, if one side of a right triangle is 0, then the hypotenuse squared is equal to the remaining side squared.

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