Answer
$a=1$
$b=-2$
$c=3$
Work Step by Step
Given Equation,
$y=ax^{2}+bx+c$
Given ordered pair solutions
$(1,2), (2,3),(-1,6)$
Substituting each ordered pair solution into the equation,
$y=ax^{2}+bx+c$
$2=a(1)^{2}+b(1)+c$
$a+b+c=2$ Equation $(1)$
$y=ax^{2}+bx+c$
$3=a(2)^{2}+b(2)+c$
$4a+2b+c=3$ Equation $(2)$
$y=ax^{2}+bx+c$
$6=a(-1)^{2}+b(-1)+c$
$a-b+c=6$ Equation $(3)$
Adding Equation $(1)$ and Equation $(3)$
$a+b+c+a-b+c=2+6$
$2a+2c=8$
$a+c=4$ Equation $(4)$
Substituting Equation $(4)$ in Equation $(1)$
$a+b+c=2$
$b+4=2$
$b=-2$
Subtracting Equation $(3)$ From Equation $(2)$
$4a+2b+c-(a-b+c)=3-6$
$4a+2b+c-a+b-c=-3$
$3a+3b=-3$
$a+b=-1$
Substituting $b$ value
$a-2=-1$
$a=-1+2$
$a=1$
Substituting $a$ and $b$ values in Equation $(4)$
$a+c=4$
$1+c=4$
$c=3$