x^2+13x+15=0
Recall (x+a)^2=x^2+2ax+a^2
Write x^2+2*(13/2)x+15, and compare the terms in x.
a=13/2
x^2+13x+15=x^2+2(13/2)x
+(13/2)^2-(13/2)^2+15
The first three terms are the expansion of
(x+13/2)^2
You expression is equivalent to
(x+13/2)^2-[(13/2)^2-15]=(x+13/2)^2-[SQRT(109)/2]^2
Use the identity for the difference of two squares a^2-b^2=(a-b)(a+b)
This gives
x^2+13x+15=(x+13/2)^2-[SQRT(109)/2]^2=
[x+13/2 -SQRT(109)/2][x+13/2+SQRT(109)/2]
The polynomial can be written as (x-a)(x-b) (Note a has nothing to do with the one used above to complete the square. You can use any name for the roots, x_1, x_2, anything.
The two roots are on the screen captures below.
PS. I prefer to manipulate exact expressions with radicals. If you do not like it use the decimal approximations.