This is equation is giving me issues $x^2 - 6x + 15 = 0 $

Question

I was given this equation $x^2 - 6x + 15 = 0 $

I tried to look for numbers whose sum is big and product of ac and i could not find any. I tried using the quadratic formula $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ and this is what i got

$x = \frac{6\pm\sqrt{-24}}{2}$

I just dont know what to do from here: any help would be appreacited.

Answer 1

So you need to find the roots as follow: Note that:

$$ i^2 = -1$$

$$ x = \frac{6\pm \sqrt{-24}}{2}=\frac{6\pm i\sqrt{24}}{2} $$

$$ x= \frac{6\pm 2i\sqrt{6}}{2} = 3\pm i\sqrt{6}$$

Further explanation ($i^2 = -1$):

let $a = \sqrt{-24}$

$a^2 = -24= 24 \times -1= 24i^2$

$a =\pm \sqrt{24} \times \sqrt{i^2}= \pm i\sqrt{24} = \pm i \times \sqrt{4} \times \sqrt{6} = \pm2i\sqrt{6}$

Answer 2

Another approach

$$x^2-6x+15=0$$

$$x^2-6x=-15$$

$$x^2-6x+9=-6$$

$$(x-3)^2=-6$$

Take the square root of both sides

$$x-3=i\sqrt6,\;x-3=-i\sqrt6$$

$$\boxed{\color{red}{x=3\pm i\sqrt6}}$$