Complete the square

A question is this type if and only if it asks to express a quadratic in the form a(x+b)²+c or similar completed square form, without further application.

43 questions · Moderate -0.9

1.02e Complete the square: quadratic polynomials and turning points
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OCR C1 2011 June Q1
4 marks Moderate -0.8
1 Express \(3 x ^ { 2 } - 18 x + 4\) in the form \(p ( x + q ) ^ { 2 } + r\).
OCR C1 2012 June Q4
6 marks Moderate -0.8
4
  1. Express \(2 x ^ { 2 } - 20 x + 49\) in the form \(p ( x - q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 20 x + 49\).
OCR C1 2016 June Q1
4 marks Easy -1.2
1
  1. Simplify \(( 2 x - 3 ) ^ { 2 } - 2 ( 3 - x ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 x ^ { 2 } - 3 x + 4 \right) \left( 5 - 2 x - x ^ { 3 } \right)\).
Edexcel Paper 1 2019 June Q5
10 marks Moderate -0.8
5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
  1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
    1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
    2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$
OCR MEI AS Paper 1 Specimen Q4
3 marks Easy -1.2
4
  1. Express \(x ^ { 2 } + 4 x + 7\) in the form \(( x + b ) ^ { 2 } + c\).
  2. Explain why the minimum point on the curve \(y = ( x + b ) ^ { 2 } + c\) occurs when \(x = - b\).
OCR MEI AS Paper 2 2024 June Q1
3 marks Easy -1.8
1 Express \(2 x ( x + 3 ) + 5 x ^ { 2 } - 2 ( x - 3 )\) in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be determined.
AQA C1 2013 January Q4
12 marks Moderate -0.8
4
    1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
    2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
    1. Find the coordinates of the vertex of the curve.
    2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
    3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).
AQA C1 2005 June Q2
10 marks Easy -1.2
2
  1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
  2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
    1. find the coordinates of the vertex (minimum point) of the curve;
    2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
    3. state the equation of the line of symmetry of the curve.
  3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).
AQA C1 2016 June Q3
6 marks Moderate -0.8
3
    1. Express \(x ^ { 2 } - 7 x + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    2. Hence write down the minimum value of \(x ^ { 2 } - 7 x + 2\).
  2. Describe the geometrical transformation which maps the graph of \(y = x ^ { 2 } - 7 x + 2\) onto the graph of \(y = ( x - 4 ) ^ { 2 }\).
    [0pt] [3 marks]
Edexcel C1 Q6
10 marks Moderate -0.8
6. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$
  1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
  2. State the maximum value of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
  4. Sketch the curve \(y = \mathrm { f } ( x )\).
AQA C1 2008 June Q4
6 marks Moderate -0.8
4
  1. Express \(x ^ { 2 } - 3 x + 4\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
    (2 marks)
  2. Hence write down the minimum value of the expression \(x ^ { 2 } - 3 x + 4\).
  3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 3 x + 4\).
CAIE P1 2014 November Q3
6 marks Moderate -0.8
  1. Express \(9x^2 - 12x + 5\) in the form \((ax + b)^2 + c\). [3]
  2. Determine whether \(3x^3 - 6x^2 + 5x - 12\) is an increasing function, a decreasing function or neither. [3]
CAIE P3 2018 June Q1
3 marks Easy -1.2
Express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
OCR C1 2014 June Q1
4 marks Moderate -0.8
Express \(5x^2 + 10x + 2\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [4]
OCR MEI C1 2010 June Q8
4 marks Moderate -0.8
Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]
OCR MEI C1 Q5
4 marks Moderate -0.5
Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]
OCR MEI C1 Q9
3 marks Moderate -0.5
Expand and simplify \((n + 2)^3 - n^3\). [3]
SPS SPS SM 2024 October Q2
5 marks Moderate -0.8
  1. Write \(3x^2 + 24x + 5\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
The finite region R is enclosed by the curve \(y = 3x^2 + 24x + 5\) and the \(x\)-axis.
  1. State the inequalities that define R, including its boundaries. [2]