Question

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

Answer 1

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

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:

`√(a^2)=√(x^2)`

Simplify:

`a=x`

And for b:

`b^2=-4`

Take the square root of both sides:

`√(b^2)=√(-4)`

Simplify:

`b=√(-4)`

Simplify the negative root using i:

`b=√(-4)=√(-1\cdot4)=√(-1)\cdot√(4)=i√(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)`