Difference of squares with coefficients, step by step

9x² − 64 is still one square minus another — but now the square in front matters.

The question

100x² − 81

100x² = (10x)²81 = 9²

(10x + 9)(10x − 9)

100x² − 81=(10x + 9)(10x − 9)

Before factoring anything, check the shape. There are two parts with a − between them, and each part has to be a perfect square: 100x² is 10x squared, and 81 is 9 squared. Both are, and it is a minus — so this is a difference of two squares.
That shape always factors the same way, and there is nothing to search for: it is the SUM times the DIFFERENCE of the two things being squared — (10x + 9)(10x − 9).
So the missing number is 10, not 100: 100x² is 10x squared, so the two brackets start 10x rather than 100x. The square root of 100x² is 10x — the root of the number AND the root of x², taken together.

How it works.

01
Before factoring anything, check the shape. There are two parts with a − between them, and each part has to be a perfect square: 100x² is 10x squared, and 81 is 9 squared. Both are, and it is a minus — so this is a difference of two squares.
02
That shape always factors the same way, and there is nothing to search for: it is the SUM times the DIFFERENCE of the two things being squared — (10x + 9)(10x − 9).
03
So the missing number is 10, not 100: 100x² is 10x squared, so the two brackets start 10x rather than 100x. The square root of 100x² is 10x — the root of the number AND the root of x², taken together.