Prove quadratic always positive/negative

A question is this type if and only if it asks to show that a quadratic expression is always positive or always negative for all real x, typically using completed square form.

3 questions · Moderate -0.5

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OCR MEI C1 Q3
12 marks Moderate -0.3
  1. Find the set of values of \(k\) for which the line \(y = 2x + k\) intersects the curve \(y = 3x^2 + 12x + 13\) at two distinct points. [5]
  2. Express \(3x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). Hence show that the curve \(y = 3x^2 + 12x + 13\) lies completely above the \(x\)-axis. [5]
  3. Find the value of \(k\) for which the line \(y = 2x + k\) passes through the minimum point of the curve \(y = 3x^2 + 12x + 13\). [2]
OCR MEI C1 Q1
11 marks Moderate -0.8
  1. Find algebraically the coordinates of the points of intersection of the curve \(y = 3x^2 + 6x + 10\) and the line \(y = 2 - 4x\). [5]
  2. Write \(3x^2 + 6x + 10\) in the form \(a(x + b)^2 + c\). [4]
  3. Hence or otherwise, show that the graph of \(y = 3x^2 + 6x + 10\) is always above the \(x\)-axis. [2]
OCR MEI C1 Q5
12 marks Moderate -0.3
  1. Write \(x^2 - 5x + 8\) in the form \((x - a)^2 + b\) and hence show that \(x^2 - 5x + 8 > 0\) for all values of \(x\). [4]
  2. Sketch the graph of \(y = x^2 - 5x + 8\), showing the coordinates of the turning point. [3]
  3. Find the set of values of \(x\) for which \(x^2 - 5x + 8 > 14\). [3]
  4. If \(f(x) = x^2 - 5x + 8\), does the graph of \(y = f(x) - 10\) cross the \(x\)-axis? Show how you decide. [2]