Quadratic with equal roots

A question is this type if and only if it specifically asks to find a parameter value for which a quadratic equation has equal (repeated) roots.

9 questions · Moderate -0.8

1.02f Solve quadratic equations: including in a function of unknown
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OCR C1 2011 January Q7
11 marks Moderate -0.3
7
  1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
  2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
  3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).
Edexcel C1 Q2
6 marks Moderate -0.3
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q30
7 marks Moderate -0.8
  1. Solve the equation \(4x^2 + 12x = 0\). [3]
\(f(x) = 4x^2 + 12x + c\), where \(c\) is a constant.
  1. Given that \(f(x) = 0\) has equal roots, find the value of \(c\) and hence solve \(f(x) = 0\). [4]
Edexcel C1 Specimen Q8
11 marks Easy -1.2
Given that $$x^2 + 10x + 36 = (x + a)^2 + b$$ where \(a\) and \(b\) are constants,
  1. find the value of \(a\) and the value of \(b\). [3]
  2. Hence show that the equation \(x^2 + 10x + 36 = 0\) has no real roots. [2]
The equation \(x^2 + 10x + k = 0\) has equal roots.
  1. Find the value of \(k\). [2]
  2. For this value of \(k\), sketch the graph of \(y = x^2 + 10x + k\), showing the coordinates of any points at which the graph meets the coordinate axes. [4]
Edexcel C1 Q3
6 marks Moderate -0.8
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q4
6 marks Moderate -0.3
  1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{(k^2 - c)}\). [4]
The equation \(x^2 + 2kx + 81 = 0\) has equal roots.
  1. Find the possible values of \(k\). [2]
Edexcel C1 Q4
6 marks Moderate -0.8
  1. Find the value of the constant \(k\) such that the equation $$x^2 - 6x + k = 0$$ has equal roots. [2]
  2. Solve the inequality $$2x^2 - 9x + 4 < 0.$$ [4]
OCR H240/02 2023 June Q1
5 marks Easy -1.3
    1. Express \(x^2 - 8x + 11\) in the form \((x - a)^2 + b\) where \(a\) and \(b\) are constants. [2]
    2. Hence write down the minimum value of \(x^2 - 8x + 11\). [1]
  2. Determine the value of the constant \(k\) for which the equation \(x^2 - 8x + 11 = k\) has two equal roots. [2]
OCR PURE Q2
4 marks Easy -1.2
  1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
  2. Solve the inequality \(6 + x - x^2 > 0\). [2]