Question

Factorise 9w² - 100


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Answer 1

`9\, w^(2) - 100 = (3\, w - 10) \, (3\, w + 10)`.

Explanation:

Fact:

`\begin{aligned} & (a - b)\, (a + b)\\ =\; & a^(2) + a\, b - a\, b - b^(2) \\ =\; & a^(2) - b^(2) \end{aligned}`.

In other words,
`(a^(2) - b^(2))`, the difference of two squares in the form
`a^(2)` and
`b^(2)`, could be factorized into
`(a - b)\, (a + b)`.

In this question, the expression
`(9\, w^(2) - 100)` is the difference between two terms:
`9\, w^(2)` and
`100`.

• 
  `9\, w^(2)` is the square of
  `3\, w`. That is:
  `(3\, w)^(2) = 9\, w^(2)`.
• On the other hand,
  `10^(2) = 100`.

Hence:

`9\, w^(2) - 100 = (3\, w)^(2) - (10)^(2)`.

Apply the fact that
`a^(2) - b^(2) = (a - b) \, (a + b)` to factorize this expression. (In this case,
`a = 3\, w` whereas
`b = 10`.)

`\begin{aligned}& 9\, w^(2) - 100 \\ =\; & (3\, w)^(2) - (10)^(2) \\ = \; & (3\, w - 10)\, (3\, w + 10)\end{aligned}`.