Question 6 - A-Level Maths

1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
  (1 mark)
The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
1. Expand \(( x - 5 ) ^ { 2 }\).
2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.