Question

Write the product as linear factors: (x^2-4(x^2-9) Without coding

Answer 1

To write the expression \(x^2 - 4(x^2 - 9)\) as linear factors, we first simplify it:

1. Start by distributing the \(-4\) across \((x^2 - 9)\):
\[
-4(x^2 - 9) = -4x^2 + 36
\]

2. Now, substitute this back into the expression:
\[
x^2 - 4(x^2 - 9) = x^2 - 4x^2 + 36
\]

3. Combine like terms:
\[
x^2 - 4x^2 = -3x^2
\]
So we have:
\[
-3x^2 + 36
\]

4. Factor out \(-3\):
\[
-3(x^2 - 12)
\]

5. We can further factor \(x^2 - 12\) by recognizing it as a difference of squares:
\[
x^2 - 12 = (x - \sqrt{12})(x + \sqrt{12})
\]
where \(\sqrt{12} = 2\sqrt{3}\), so we can rewrite it as:
\[
x^2 - 12 = (x - 2\sqrt{3})(x + 2\sqrt{3})
\]

6. Therefore, putting everything together, we have:
\[
-3(x - 2\sqrt{3})(x + 2\sqrt{3})
\]

Thus, the original expression \(x^2 - 4(x^2 - 9)\) as linear factors is:
\[
-3(x - 2\sqrt{3})(x + 2\sqrt{3})
\]