Quadratic functions (Grade 11)

Was the question actually written that way - its very imprecise and technically ambiguous, because the roots are one thing but saying a, b, are the roots of (x - 1)^2 = f(k) is just gross. Let's assume they mean (x - 1)^2 - f(k) is a quadratic of x and k is a fixed real number.

Let k^2 - k + 2 be a constant say c

x^2 -2x - (1+c) is your quadratic.

roots are [2 +/- (4 + 4(1 + c))^1/2]/2.

They want you to answer what is abs(a - b) minimum value,

So what is the minimum of (reducing and eliminating terms) (8 + 4c)^1/2 or (4k^2 - 4k + 16)^1/2

Which has a minimum value dependent k in the function (4(k^2 - k + 4))^1/2 = 2(k^2 - k + 4)^1/2 (i.e. when f'(k) = 0) so dervive by k to get

f'(k^2 - k + 4) = 2k - 1 = 0 so k = 1/2

so you answer is min a-b = 2(1/4 + -1/2 + 4)^1/2 = 2(3.75)^1/2 = 3.8729833462074168851792653997824 (root 15)

* * *

Now is your Sum of roots (a + b) = 2 ??? did you calcuate that in some way unmentioned or is that an extra fact you choose to add late rather than at the beginning? Let's treat it as an extra fact you introduced late. What it basically tells us is

[2 + (4 + 4(1 + c))^1/2]/2 + [2 - (4 + 4(1 + c))^1/2]/2 =
2/2 + 2/2 + [(4 + 4(1 + c))^1/2]/2 - [(4 + 4(1 + c))^1/2]/2 =
1 + 1 = 2 ... so no new information there!

This is just the whole question:

Let a and b be the roots of the equation (x-1)^2 = k^2 - k + 2
where k is a real number
(1) Show that a and b are real and distinct
(2) Find the minimum value of abs(a-b)

The others are my own calculation.
I think the question wants us to treat this as a quadratic of x with k being a constant.
I have done (1) by proving the discriminant > 0. So I asked for solution of (2), which I think you have already showed me the steps I need.
Thank you very much.

(x-1)^2=k^2-k+2
let m^2=k^2-k+2
(x-1)^2=m^2
x^2-2x+(1-m^2)=0
so
a,b=1+/-m
so a+b has to be 1+m+1-m=2
|a-b|=|1+m-(1-m)|=|2m|=|2sqrt(k^2-k+2)|
d|a-b|/dk=(-1/2)2[sqrt(k^2-k+2)]^(-1/2)(2k-1)=-(2k-1)/sqrt(k^2-k+2)=0
so 2k-1=0 iff (k^2-k+2)<>0
k=1/2 k^2-k+1=1/4-1/2+2=1 ¾
so k=1/2 to minimize |a-b|

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