what happens to the middle term when you factor a perfect square?

Question

like, the answer to a^2+10ab+25b^2 is (a+5b)^2. What happened to 10ab?

Answer 1

To understand the process, work backwards...

(a+5b)^2 = (a+5b)(a+5b)
= a^2 + 5ab + 5ab + 25b^2
= a^2 + 10ab + 25b^2

Answer 2

We know (A + B)^2 = A^2 +2.A.B + B^2
Given: a^2 + 10ab + 25b^2 = a^2 + 2.a.5b + (5b)^2
Here, A = a, B = 5b
So a^2 + 10ab + 25b^2 = (a + 5b)^2

Another method:
a^2 + 10ab + 25b^2
= a^2 + 5ab + 5ab + 25b^2 (break the middle term such that 5+5 = 10 = middle term and 5x5 = 25 = last term x first term = 25x1)
= a(a + 5b) + 5b(a + 5b)
= (a + 5b)(a + 5b)
= (a + 5b)^2

Note: If a binomial function is squared, 4 terms are produced, of which two terms are algebraically additive which gives us a TRINOMIAL.
When we factorize a TRINOMIAL, the middle term is broken into two terms appropriately and then factorize.

Answer 3

You have expressed the middle term as a sum of two other terms, which enabled you to pair them, and factor to the final result.

Check it here:
a^2+10ab+25b^2 = (to factor this out, we will have to have something that will enable us to extract 5b from 25b^2, and a from a^2; "it" has to form a sum equal to 10ab: we need 10ab = something*ab + somethingelse*ab; other times it might not be expressed as two identical terms)
= a^2 + 5ab + 5ab + 25b^2 (this time, 5ab + 5ab might match our needs; if the middle term was negative, you would need a difference, e.g. -5ab-5ab)
= a(a+5b) + 5b(a+5b)
= (a+5b)(a+5b)
= (a+5b)^2

Answer 4

10ab is used to check if your chose number is right or not.
in example above you wrote (a+5b)^2, it means (a+5b)*(a+5b), why did you put 5b and 5b in other parenthesis? you could put (a+b)(a+25b) in other case, but because your number is 10ab, the possible number is 5b and 5b, if you multiply the inside numbers (5b*a) then add with the outside number (a*5b) you will get 10ab.

hope it helps.

Answer 5

(a+5b)^2 = (a+5b)(a+5b) = a^2 + 5ab + 5ab + 25b^2 = a^2 + 10ab + 25b^2

Nothing has happened to it.

Answer 6

Just remember that you can't square a sum by squaring each part of the sum; you have to write it out and use FOIL.

(3 + 4)^2 is not 3^2 + 4^2

49 is not 9 + 16

Answer 7

10ab = 2*a*5b
2 = first term
5b = second term
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