Question

Use complex conjugates to factor the expression $9x^2+36$.

Answer 1

Answer:

$9x^2+36=9(x+2i)(x-2i)$

Step-by-step explanation:

We have the expression:

$9x^2+36$

We can factor this using complex conjugates. Essentially, we will use the difference of two squares:

$a^2-b^2=(a-b)(a+b)$

First, we can factor out a 9 from our expression. This gives us:

$9(x^2+4)$

We can now rewrite our expression as:

$9(x^2-(-4))$

Therefore, our a² is $x^2$ and our b² is -4.

Let’s solve for each of them individually. So, for a:

$a^2=x^2$

Take the square root of both sides:

$\sqrt{a^2}=\sqrt{x^2}$

Simplify:

$a=x$

And for b:

$b^2=-4$

Take the square root of both sides:

$\sqrt{b^2}=\sqrt{-4}$

Simplify:

$b=\sqrt{-4}$

Simplify the negative root using i:

$b=\sqrt{-4}=\sqrt{-1\cdot4}=\sqrt{-1}\cdot\sqrt{4}=i\sqrt{4}=2i$

Therefore, we have $a=x$ and $b=2i$.

So, by using the difference of two squares:

$9x^2+36=9(x+2i)(x-2i)$