Given: #" "x^2-2x+1=18#
Subtract 18 from both sides giving:
#" "x^2-2x-17=0#.........................(1)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This one is more straight forward as the coefficient of #x^2# is 1
#color(blue)("Step 1")#
Write as:
#(x^2-2x)-17=0#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 2")#
#color(brown)("At this point we will change the intrinsic value of the equation so it")# #color(brown)("will need to have a correction added later.")#
Take the power to be outside the brackets
#(x-2x)^(color(red)(2))-17#
Remove the #x# from the #-2x# leaving just the #color(red)(-2)#
#(xcolor(red)(-2))^2-17#
halve the -2 that is inside the brackets so that you have #color(red)(-1)#
#(xcolor(red)(-1))^2-17#
#color(brown)("Now we add the correction")#
#color(green)("This takes the equation back to its original value so we can once again equate it to zero")#
Let #color(red)(k)# be some constant
#(x-1)^2color(red)(+k)-17=0# ................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 3")#
#color(brown)("By expanding the brackets and comparing to the original equation")##color(brown)("we can determine the appropriate value of "k)#
Expanding equation (2)
#cancel(x^2)-cancel(2x)+1+k-cancel(17) " "=" "cancel(x^2)-cancel(2x)-cancel(17)#
#k=-1# so equation (2) becomes:
#color(blue)((x-1)^2-18=0)# .................................(3)
By comparing the values in the vertex format equation you can see how to obtain the vertex coordinates.
Multiply the constant inside the bracket by #(-1)# to get #x_("vertex")# giving 1 and #y_("vertex")# can be read directly as -18
#color(blue)("vertex "-> (x,y)" "->" "(1,-18))#
;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 4")#
y-intercept at #x=0#
#=>(x-1)^2-18=y" "->" "(0-1)^2-18=y#
#color(blue)(y_("intercept")= -17)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Step 5")#
x-intercepts when y=0 so from equation (3)
#sqrt( (x-1)^2)=sqrt(18)#
#x-1=+-3sqrt(2)#
#x=1+-3sqrt(2)#
#color(blue)(x_("intercept")~= + 5.24" "or" "-3.24# to 2 decimal places
