FIRST METHOD:
The zeros of the given polynomial is determined by completing the square and applying polynomial properties
color(brown)((a-b)^2=a^2-2ab+b^2)
color(violet)(a^2-b^2=(a-b)(a+b))
-2x^2+16x-26
=color(blue)(-2)(color(blue)1x^2color(blue)(-8)xcolor(blue)(+13))
=-2(x^2-8x+13color(red)(+3-3))
=-2(x^2-8x+16-3)
=-2(color(brown)(x^2-2(4)x+4^2)-3)
=-2(color(brown)((x-4)^2)-3)^
=-2color(violet)(((x-4)^2-(sqrt3)^2))
=-2color(violet)(((x-4)-sqrt3)((x-4)+sqrt3))
=-2((x-(4+sqrt3)((x-(4-sqrt3))
The zeros of the given algebraic expression is
-2x^2+16x-26=0
-2((x-(4+sqrt3)((x-(4-sqrt3))=0
x-(4+sqrt3)=0rArrx=4+sqrt3
Or
x-(4-sqrt3)=0rArrx=4-sqrt3
SECOND METHOD:
The zeros of the given polynomial is determined by using the quadratic formula
-2x^2+16x-26
color(blue)(delta=b^2-4ac)=16^2-4(-2)(-26)
delta=256-208=48
Roots are:
color(red)(x_1=(-b-sqrtdelta)/(2a))=(-16-sqrt48)/(2(-2))=(-16-sqrt(2^4xx3))/(-4)
color(red)(x_1)=(-16-4sqrt3)/(-4)=(-4(4+sqrt3))/(-4)=(cancel(-4)(4+sqrt3))/cancel(-4)
color(red)(x_1)=4+sqrt3
color(red)(x_2=(-b+sqrtdelta)/(2a))=(-16+sqrt48)/(2(-2))=(-16+sqrt(2^4xx3))/(-4)
color(red)(x_2)=(-16+4sqrt3)/(-4)=(-4(4-sqrt3))/(-4)=(cancel(-4)(4-sqrt3))/cancel(-4)
color(red)(x_2)=4-sqrt3
