We will have to complete the square for this quadratic which will put the equation in vertex form.
First lets solve for the y variable by dividing both sides by 7
#(cancel7y)/cancel7=3/7x^2-2/7x+12/7#
Set the equation equal to zero.
#0=3/7x^2-2/7x+12/7#
Subtract #12/7# from both sides
#0color(red)(-12/7)=3/7x^2-2/7x+12/7color(red)(-12/7)#
Simplify
#color(red)(-12/7)=3/7x^2-2/7x#
Factor out #3/7#
#-12/7=3/7(x^2-2/cancel7(cancel7/3)x)#
Simplify
#-12/7=3/7(x^2-2/3x)#
Take the coefficient of x and divide it by 2 and then square it
#((-2/3)/2)^2=(-2/3*1/2)^2=(-2/6)^2=(-1/3)^2=1/9#
Add #1/9# to the right side and add #3/7(1/9)# to the left side because we factored out #3/7# in the beginning. This process will keep the equation balanced.
#color(red)(3/7(1/9))-12/7=3/7(x^2-2/3x+color(red)(1/9))#
Simply
#color(red)(cancel3/7(1/(cancel9 3)))-12/7=3/7(x^2-2/3x+color(red)(1/9))#
#1/21-12/7=3/7(x^2-2/3x+color(red)(1/9))#
Find Common Denominator
#1/21-12/7*color(red)(3/3)=3/7(x^2-2/3x+color(red)(1/9))#
#1/21-36/21=3/7(x^2-2/3x+color(red)(1/9))#
The right side is a perfect square trinomial
#1/21-36/21=3/7(x-1/3)^2#
#-35/21=3/7(x-1/3)^2#
#-(cancel35 5)/(cancel 21 3)=3/7(x-1/3)^2#
#-5/3=3/7(x-1/3)^2#
Add #5/3# from both sides
#color(red)(5/3)-5/3=3/7(x-1/3)^2color(red)(+5/3)#
#0=3/7(x-1/3)^2color(red)(+5/3)#
Vertex form #=> y=(x-h)^2+k#
Vertex #=> (h,k) => (1/3,5/3)#
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