Question

How do I simplify this imaginary expression on the left?

`x^2-(6+3i)x +k=0`

one solution is 3

I solved for k

which is `9-9i` (answer is correct)

But I am having issues factoring and simplifying when we plug in k in the original

`x^2-(6+3i)x + (9-9i)=0`

Answer 1

`k = 9 + 9i`

Explanation:

Given:

`x^2-(6+3i)x+k = 0" "` with root `3`

Putting `x=3` into the equation, we get:

`0 = color(blue)(3)^2-(6+3i)(color(blue)(3))+k`

`color(white)(0) = 9-18-9i+k`

`color(white)(0) = -9-9i+k`

So:

`k = 9 + 9i`

Our original equation becomes:

`x^2-(6+3i)x+(9+9i) = 0`

Note that:

`6 + 3i = 3 + (3+3i)" "` which is the sum of the roots

`9 + 9i = 3(3+3i)" "` which is the product of the roots

Factoring, we have:

`x^2-(6+3i)x+(9+9i) = (x-3)(x-(3+3i))`