1.02e Complete the square: quadratic polynomials and turning points

Edexcel C1 Q4

7 marks Moderate -0.8

\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]

Given that \(k = 4\),

1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]

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

8 marks Moderate -0.8

1. By completing the square, find in terms of \(k\) the roots of the equation $$x^2 + 2kx - 7 = 0.$$ [4]
2. Prove that, for all values of \(k\), the roots of \(x^2 + 2kx - 7 = 0\) are real and different. [2]
3. Given that \(k = \sqrt{2}\), find the exact roots of the equation. [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]

OCR C1 2006 June Q3

7 marks Moderate -0.8

1. Express \(2x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). [4]
2. Solve \(2x^2 + 12x + 13 = 0\), giving your answers in simplified surd form. [3]

OCR C1 2013 June Q4

7 marks Moderate -0.8

1. Express \(3x^2 + 9x + 10\) in the form \(3(x + p)^2 + q\). [3]
2. State the coordinates of the minimum point of the curve \(y = 3x^2 + 9x + 10\). [2]
3. Calculate the discriminant of \(3x^2 + 9x + 10\). [2]

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 Q7

5 marks Easy -1.2

Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]

OCR MEI C1 2006 January Q11

13 marks Moderate -0.8

1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]

OCR MEI C1 2009 June Q9

5 marks Easy -1.2

1. Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\). [3]
2. Write down the coordinates of the minimum point on the graph of \(y = x^2 + 6x + 5\). [2]

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 2011 June Q11

11 marks Moderate -0.8

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
3. For the curve \(y = 4x^2 + 24x + 31\),
   1. write down the equation of the line of symmetry, [1]
   2. write down the minimum \(y\)-value on the curve. [1]

OCR MEI C1 2013 June Q8

5 marks Moderate -0.8

Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]

Edexcel C1 Q6

8 marks Moderate -0.3

1. By completing the square, find in terms of the constant \(k\) the roots of the equation $$x^2 + 4kx - k = 0.$$ [4]
2. Hence find the set of values of \(k\) for which the equation has no real roots. [4]

Edexcel C1 Q6

8 marks Moderate -0.8

$$\text{f}(x) = 2x^2 - 4x + 1.$$

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\text{f}(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = \text{f}(x)\). [1]
3. Solve the equation \(\text{f}(x) = 3\), giving your answers in exact form. [3]

Edexcel C1 Q2

4 marks Easy -1.2

1. Express \(x^2 + 6x + 7\) in the form \((x + a)^2 + b\). [3]
2. State the coordinates of the minimum point of the curve \(y = x^2 + 6x + 7\). [1]

Edexcel C1 Q5

6 marks Moderate -0.3

1. By completing the square, find in terms of the constant \(k\) the roots of the equation $$x^2 + 2kx + 4 = 0.$$ [4]
2. Hence find the exact roots of the equation $$x^2 + 6x + 4 = 0.$$ [2]

OCR C1 Q3

6 marks Moderate -0.8

1. Express \(x^2 - 10x + 27\) in the form \((x + p)^2 + q\). [3]
2. Sketch the curve with equation \(y = x^2 - 10x + 27\), showing on your sketch
   1. the coordinates of the vertex of the curve,
   2. the coordinates of any points where the curve meets the coordinate axes. [3]

OCR C1 Q6

8 marks Moderate -0.8

\(f(x) = 2x^2 - 4x + 1\).

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$f(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = f(x)\). [1]
3. Solve the equation \(f(x) = 3\), giving your answers in exact form. [3]

OCR C1 Q7

11 marks Moderate -0.8

The curve \(C\) has the equation \(y = x^2 + 2x + 4\).

1. Express \(x^2 + 2x + 4\) in the form \((x + p)^2 + q\) and hence state the coordinates of the minimum point of \(C\). [4]

The straight line \(l\) has the equation \(x + y = 8\).

1. Sketch \(l\) and \(C\) on the same set of axes. [3]
2. Find the coordinates of the points where \(l\) and \(C\) intersect. [4]

OCR C1 Q5

8 marks Moderate -0.3

$$f(x) = x^2 - 10x + 17.$$

1. Express \(f(x)\) in the form \(a(x + b)^2 + c\). [3]
2. State the coordinates of the minimum point of the curve \(y = f(x)\). [1]
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(y = f(x) + 4\), [2]
   2. \(y = f(2x)\). [2]

OCR C1 Q8

12 marks Moderate -0.3