Question

What is the difference of squares formula and how to apply it to factor 200x^2-18?

Answer 1

The difference of squares formula states that for any two expressions \( a \) and \( b \):

\[
a^2 - b^2 = (a - b)(a + b)
\]

To apply this to factor \( 200x^2 - 18 \), we first need to express it in the form of a difference of squares.

1. **Factor out a common factor (if applicable)**:
Notice that both terms in \( 200x^2 - 18 \) have a common factor. We can factor out 2 from the expression:

\[
200x^2 - 18 = 2(100x^2 - 9)
\]

2. **Identify the squares**:
Now, we look at the expression inside the parentheses, \( 100x^2 - 9 \). We can see that:

\[
100x^2 = (10x)^2 \quad \text{and} \quad 9 = 3^2
\]

Thus, we can rewrite \( 100x^2 - 9 \) as:

\[
100x^2 - 9 = (10x)^2 - 3^2
\]

3. **Apply the difference of squares formula**:
Now we can apply the difference of squares formula:

\[
(10x)^2 - 3^2 = (10x - 3)(10x + 3)
\]

4. **Combine back with the common factor**:
Substituting this back into the factored expression gives us:

\[
200x^2 - 18 = 2((10x - 3)(10x + 3))
\]

Thus, the fully factored form of \( 200x^2 - 18 \) is:

\[
200x^2 - 18 = 2(10x - 3)(10x + 3)
\]