I'm finding these really hard, please could someone attempt these, many thanks! Especially explaining why real values of x you state apply.
Q1) The equation of a curve is y=ax^2 - 2bx + c, where a, b and c are constants with a > 0.
a) Find, in terms of a, b and c, the coordinates of the vertex of the curve.
b) Given that the vertex of the curve lies on the line y=x, find an expression for c in terms of a and b. Show that in this case, whatever the value of b, c >= -1/4a (>= means greater than or equal to).
For this question, I found answers to part a), but am stuck on b). I know part a) answer is in co-ordinates (b/a , c-(b^2/a)). I also know some of part b), c is [b(b+1)/a], so I basically don't get why the show that thing is that.
Q2) a) Express 9x^2 +12x + 7 in the form (ax + b)^2 + c where a, b and c are constants whose values are to be found.
b) Find the set of values taken by: 1/(9x^2 +12x + 7) for real values of x.
For this question, I could get a) with a = 3, b=2, c=3. But have no idea what "real values of x" mean and please provide reasons why.
Q3) a) Express 9x^2 - 36x + 52 in the form (Ax-B)^2 + C, where A, B and C are integers. Hence, or otherwise, find the set of values taken by 9x^2 - 36x + 52 for real x. Again I don't know what "real x" means, nor how to work out the answer.
Many thanks in advance, this is too tricky for me ;(
