Understanding Algebra: The Mystery of the Middle Term Explained

[Rob D]

Please forgive this most basic and fundamental of algebra questions but as I enter the calculus arena there is one small algebra function that has eluded intuitive understanding. In factoring quadratics and other polynomials, I am very successfully able to work the operations of factoring but one thing bugs me. I'm a self-educating 64 yo physics student so I have no teacher or colleague to ask.

Where did the middle term come from?
If I factor: 4x² - 28x + 49 I get (2x-7)(2x-7) or (2x-7)²

If I then "FOIL" the terms (2x)(2x)-14x-14x +(-7x)(-7x) I again get 4x²-28x+49

Intuitively, at least for my old brain, (2x-7)² gives 4x²+49

Just writing this down I'm feeling dumb but using this method how or why is the 28x generated from (2x-7)²?

Many thanks and Sorry for the High School question.

 

[Borek]

Intuitively, at least for my old brain, (2x-7)² gives 4x²+49

Your intuition is wrong.

$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a-b)^2 = a^2 - 2ab + b^2$$

$$(a+b)(a-b) = a^2 - b^2$$

(This is not a rocket science - you should be able to derive all three in no time).

 

[Rob D]

You intuition is wrong.

$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a-b)^2 = a^2 - 2ab + b^2$$

$$(a+b)(a-b) = a^2 - b^2$$

(This is not a rocket science - you should be able to derive all three in no time).

Borek,

Spaciba Bolshoi, for the response.

Actually I know my intuition is flawed. I am well past this level of study but I've always had a niggling problem with (a+b)² not simply being (a+b)(a+b). While I can do the math, I've never understood how the 2ab was generated.

Since I can do the math, perhaps I should just work the solutions and keep my trap shut but I want to understand.

I'm not dumb, really, ask my wife.

 

[Borek]

Try to derive these formulas.

$$(a+b)^2 = (a+b)(a+b) = \dots$$

 

[Rob D]

Try to derive these formulas.

$$(a+b)^2 = (a+b)(a+b) = \dots$$

(a)(a)+ab+ab+(b)(b) = a²+2ab+b²

Yeah, of course I know you're right but it bugs me. But sometimes I wonder if the outcome is not more a product of procedure and dogma than true mathematics.

But, as I always say "Cooperate and Graduate."

 

[eggshell]

Please forgive this most basic and fundamental of algebra questions but as I enter the calculus arena there is one small algebra function that has eluded intuitive understanding. In factoring quadratics and other polynomials, I am very successfully able to work the operations of factoring but one thing bugs me. I'm a self-educating 64 yo physics student so I have no teacher or colleague to ask.

Where did the middle term come from?
If I factor: 4x² - 28x + 49 I get (2x-7)(2x-7) or (2x-7)²

If I then "FOIL" the terms (2x)(2x)-14x-14x +(-7x)(-7x) I again get 4x²-28x+49

Intuitively, at least for my old brain, (2x-7)² gives 4x²+49

Just writing this down I'm feeling dumb but using this method how or why is the 28x generated from (2x-7)²?

Many thanks and Sorry for the High School question.

(2x-7)(2x-7)

First=2x(2x)= 4x^2
Outer=2x(-7)=-14x
Inner=(-7)(2x)=-14x
Last= (-7)(-7)= 49

4x^2-28x+49

all you're doing is using the distributive property, there is no reason as to why you must FOIL, other than that the acronym makes it easier for people to remember. if you take 2x from the first parentheses and distribute it to the second parentheses, and then repeat the process for the second term in the first parentheses, then add your two results, you will get the same answer.

2x(2x)+2x(-7)
+
-7(2x)-(7)(-7)

 

[SammyS]

Borek,

Spaciba Bolshoi, for the response.

Actually I know my intuition is flawed. I am well past this level of study but I've always had a niggling problem with (a+b)² not simply being (a+b)(a+b). While I can do the math, I've never understood how the 2ab was generated.

Since I can do the math, perhaps I should just work the solutions and keep my trap shut but I want to understand.

I'm not dumb, really, ask my wife.

Look at this geometrically.

 

[Integral]

To develop your intuition on this just but in numbers.

(1 + 2)²= 3²=9
(1+2)*(1+2)=3*3=9
(1)²+(2)²= 1+ 4 = 5
Clearing you need more then just the squares of the 2 terms.

Just doing the polynomial multiplication should lead to better insight also.

 

[SammyS]

Look at this geometrically.

There is a graphic in the Wikipedia article on Factorization which shows the relationship between a², b², and (a + b)². http://en.wikipedia.org/wiki/Perfect_square_trinomials#Perfect_square_trinomials

Here is a copy of that graphic:

 

[Rob D]

Thanks to all who responded. This horse is starting to smell and I'm tired of beating it so so please accept my gratitude for the insight.
RD