Roots given, find equation constants

A question is this type if and only if the roots of a quadratic are given explicitly and the task is to find the values of constants (p, q, a, b) in the equation, typically using sum and product of roots or substitution.

2 questions · Moderate -0.8

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CAIE P1 2011 June Q10
10 marks Moderate -0.8
10
  1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
  2. Find the coordinates of \(Q\).
  3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).
CAIE P1 2016 June Q6
7 marks Moderate -0.8
6
  1. Find the values of the constant \(m\) for which the line \(y = m x\) is a tangent to the curve \(y = 2 x ^ { 2 } - 4 x + 8\).
  2. The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(\mathrm { f } ( x ) = 0\) are \(x = 1\) and \(x = 9\). Find
    1. the values of \(a\) and \(b\),
    2. the coordinates of the vertex of the curve \(y = \mathrm { f } ( x )\).