Complete the square, then discriminant

A question is this type if and only if it requires first expressing a quadratic in completed-square form and then using that form (or the resulting discriminant) to determine the nature or number of roots.

2 questions · Moderate -0.6

1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points
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OCR C1 2008 June Q10
14 marks Moderate -0.8
10
  1. Express \(2 x ^ { 2 } - 6 x + 11\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 6 x + 11\).
  3. Calculate the discriminant of \(2 x ^ { 2 } - 6 x + 11\).
  4. State the number of real roots of the equation \(2 x ^ { 2 } - 6 x + 11 = 0\).
  5. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 6 x + 11\) and the line \(7 x + y = 14\).
OCR PURE Q4
6 marks Moderate -0.3
4
  1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
  3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).