color(blue)("Determine the x-intercepts")
Note that 23 is a prime number and as the coefficient of x IS NOT +1-23=-22 the roots will be fractional. Thus use the formula
Given that: y=ax^2+bx+c" where "x=(-b+-sqrt(b^2-4ac))/(2a)
In this case: a=+1"; "b=-1"; "c=-23
x=(+1+-sqrt((-1)^2-4(1)(-23)))/(2(1))
x=(1+-sqrt(93))/2
93 is not prime so we can 'hunt' for squared values as factors of it. The whole number factors turn out to be 3 and 31. Both of which are prime so we are stuck with sqrt(93) as the exact value.
Thus x=1/2+-sqrt(93)/2" "larr exact values
" "x~~-4.321825..." and "x~~5.321825...
" "x~~-4.32" and "x~~5.32 to 2 decimal places
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color(blue)("Determine the vertex")
x_("vertex") will be midpoint between the x-intercepts
x_("vertex") ->(-4.32+5.32)/2=0.5
As a check this is a 'sort of cheat method'
From y=ax^2+bx+c:" "x_("vertex")=(-1/2)xxb
" "-> (-1/2)xx(-1)=+0.5
y_("vertex")=x^2-x-23" "=" "(0.5)^2-0.5-23
" "=-23.25 =-93/4
Vertex->(x,y)=(1/2,-93/4) ->(0.5,-23.25)
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color(blue)("Determine the y-intercept")
Set x=0
y=x^2-x-23-=>y=0^2+0-23
y_("intercept")=-23
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