Complete the square technique

Express a quadratic in completed square form a(x + b)² + c, often as a step toward solving inequalities or finding ranges.

6 questions · Moderate -0.8

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CAIE P1 2010 June Q9
11 marks Moderate -0.8
9 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } - c\).
  2. State the range of f .
  3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) < 21\). The function g is defined by \(\mathrm { g } : x \mapsto 2 x + k\) for \(x \in \mathbb { R }\).
  4. Find the value of the constant \(k\) for which the equation \(\operatorname { gf } ( x ) = 0\) has two equal roots.
CAIE P1 2014 June Q2
4 marks Moderate -0.8
2
  1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
  2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).
CAIE P1 2014 June Q8
8 marks Moderate -0.8
8
  1. Express \(2 x ^ { 2 } - 10 x + 8\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and use your answer to state the minimum value of \(2 x ^ { 2 } - 10 x + 8\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } - 10 x + 8 = k x\) has no real roots.
CAIE P1 2019 June Q1
5 marks Easy -1.2
1 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 8\) for \(x \in \mathbb { R }\).
  1. Express \(x ^ { 2 } - 4 x + 8\) in the form \(( x - a ) ^ { 2 } + b\). \includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-02_67_1569_397_328}
  2. Hence find the set of values of \(x\) for which \(\mathrm { f } ( x ) < 9\), giving your answer in exact form.
CAIE P1 2002 November Q11
12 marks Moderate -0.3
11
  1. Express \(2 x ^ { 2 } + 8 x - 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. For the curve \(y = 2 x ^ { 2 } + 8 x - 10\), state the least value of \(y\) and the corresponding value of \(x\).
  3. Find the set of values of \(x\) for which \(y \geqslant 14\). Given that \(\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10\) for the domain \(x \geqslant k\),
  4. find the least value of \(k\) for which f is one-one,
  5. express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) in this case.
CAIE P1 2016 November Q1
4 marks Moderate -0.8
1
  1. Express \(x ^ { 2 } + 6 x + 2\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } + 6 x + 2 > 9\).