Simplify 4c^2 - 9?

(2c - 3)(2c + 3)

when you have a difference of squares x^2 - y^2, it can always be factored to give (x+y)(x-y).

in this case we have

x^2 = 4c^2
so x = sqrt(4c^2) = 2c

y^2 = 9
so y = sqrt(9) = 3

therefore, 4c^2 - 9 = (2c+3)(2c-3)

4c² = (2c)² = (2c)(2c)
9 = 3² = (3)(3)

4c² - 9 = (2c + 3)(2c - 3)

Listen to the 2nd answerer. This is just one of the patterns you have to learn when you are in algebra; don't be afraid of it, and realize that patterns like

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

work for ANY 'a' and 'b'...even when (in this case)

a^2 = 4c^2, so that a = +/- 2c

[remember, square roots can be either positive or negative...there is an often unspoken assumption that we take the positive root here]

SO: When you are doing algebra, don't forget that the "unknowns", the variables, can take on ANY values...including expressings with OTHER variables and constants! This is part of being able to "see" these patterns in more complex expressions...which is important, because you really only have a few patterns to memorize, but a LOT of ways that they "hide" in expressions and can be used to reduce them.

Don't be afraid of the letters! LOL

4c^2 - 9
= (2c + 3)(2c - 3)

c, 4, and 9 are square roots of c, 2, and 3.

Thus to simplify 4c^2 - 9 is rather simple.

( 2c - 3)( 2c + 3)

It must be -3 and + 3 so that there would not be any xc terms and because you must obtain -9 and not +9.

4c^2 can also be written as (2c)^2. And 9 can also be written as 3^2.

so the expression becomes

(2c)^2 - 3^2

By using the identity
a^2 - b^2 = (a+b) ( a-b)

The expression becomes
(2c + 3) (2c - 3)

Find the square root of each number. The root of 4c^2 is 2c and the root of 9 is 3. Remember the roots can be positive or negative. Now you just put it in btackets. One bracket is + and the other one is -. It doen not matter which one is first.
(2c + 3)(2c - 3)