The generalized standard form for a quadratic equation is
#color(white)("XXX")y=ax^2+bx+c#
We are told that solutions to the required quadratic are
#color(white)("XXX")(x,y) in {(-1,5),(0,3),(3,9)}#
That is
#{(5=a * (-1)^2+b * (-1) +c),
(3=a * (0)^2+b * (0)+c),
(9=a * (3)^2+b * (3) +c):}#
simplifying, we can write:
#{
([1]color(white)("XX")a-b+c=5),
([2]color(white)("XXXXXXX")c=3),
([3]color(white)("XX")9a+3b+c=9)
:}#
Using [2] we can further simplify [1] and [3] to get
#{
([1]rarr[4]color(white)("XX")a-b=2),
([2]rarr[5]color(white)("XX")9a+3b=6color(white)("XX")rarrcolor(white)("XX")3a+b=2)
:}#
Adding [4] and [5]
#{
color(white)("XXXX")[6]color(white)("XX")4a=4color(white)("XX")rarrcolor(white)("XX")a=1
:}#
Using [6], we can substitute #1# for #a# back in [4] to get
#{color(white)("XXXX")[7]color(white)("XX")1-b=2color(white)("XX")rarrcolor(white)("XX")b=-1
:}#
Therefore
#color(white)("XXX")(a,b)=(1,-1,3)#
and the required quadratic equation is
#color(white)("XXX")y=1x^2-1x+3 (=x^2-x+3)#
