we'll check for the discriminant:
[tex] \sqrt{b^2-4ac} [/tex]
a=1
b=-6
c=13
[tex] \sqrt{(-6)^2 -4(1)(13)} [/tex]
[tex] \sqrt{36-52} [/tex]
[tex] \sqrt{-16} [/tex]
take note that the number inside the radical sign is negative therefore the roots of the equation are imaginary numbers. we will also then have the factors of the equation as imaginary.
use quadratic formula given as:
[tex]x = \frac{-b(+-) \sqrt{b^2-4ac} }{2a} [/tex]
we already have the discriminant as √-16
[tex]x = \frac{-(-6)(+-) \sqrt{-16} }{2(1)} [/tex]
-16 can be factored as 16 and -1 and -1 is i²
[tex]x = \frac{6(+-) \sqrt{16i^2} }{2} [/tex]
[tex]x = \frac{6(+-)4i}{2} [/tex]
[tex]x = 3 (+-) 2i[/tex]
x = 3 + 2i, x=3 - 2i
expressing the roots as factors you'll have:
x = 3 + 2i
x -(3 + 2i) = 0
x = 3 - 2i
x - (3 - 2i) = 0
then the factors of x²-6x+13 are
[x - (3 + 2i)] and [x - (3 - 2i)]
