Math  /  Algebra

QuestionFactor the difference of two squares. 16x28116 x^{2}-81

Studdy Solution

STEP 1

What is this asking? We need to rewrite 16x28116x^2 - 81 as a product of two simpler expressions. Watch out! Remember that not all expressions can be factored as a difference of squares!
Make sure both terms are perfect squares and there's a subtraction sign between them.

STEP 2

1. Rewrite as a difference of squares
2. Factor

STEP 3

Alright, let's **rewrite** our expression to make it super clear that we're dealing with a difference of squares!
We can rewrite 16x216x^2 as (4x)2(4x)^2 since 44=164 \cdot 4 = 16 and xx=x2x \cdot x = x^2.
Similarly, we can rewrite 8181 as 929^2 since 99=819 \cdot 9 = 81.

STEP 4

So, our expression 16x28116x^2 - 81 becomes (4x)292(4x)^2 - 9^2.
See how we've got a **difference of squares** now?
It's much clearer!

STEP 5

Now, for the grand finale!
Remember the **difference of squares formula**: a2b2=(a+b)(ab)a^2 - b^2 = (a+b)(a-b).
This formula is a *superpower*!

STEP 6

In our case, aa is 4x4x and bb is 99.
So, plugging these into our **superpowered formula**, we get (4x)292=(4x+9)(4x9)(4x)^2 - 9^2 = (4x + 9)(4x - 9).

STEP 7

So, the factored form of 16x28116x^2 - 81 is (4x+9)(4x9)(4x+9)(4x-9)!
Boom!

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