Completing the Square (Edexcel GCSE Maths: Higher)

The expression  can be written in the form  for all values of .

Find the value of and the value of .

The equation of a curve is  where 

The diagram shows part of a sketch of the graph of .

The minimum point of the curve is .

Write down the coordinates of .

Given that  for all values of .

i) Find the value of  and the value of .

[2]

ii) Hence write down the coordinates of the turning point of the graph of .

[1]

Given that and are integers,

express  in the form 

Express   in the form  where and are integers.

Write in the form 

The line  is the line of symmetry of the curve with equation 

Using your answer to part (a) or otherwise, write down an equation of .

Express in the form

Show your working clearly.

The equation of a curve is

By completing the square, work out the coordinates of the turning point.

You must show your working.

i) Write in the form .

[3]

ii) Solve the equation .
Give your answers in surd form as simply as possible.

= ........... or = .......... [4]

Write    in the form  

Sketch the graph of  ,  indicating the coordinates of the turning point.

i) Write  in the form .  

[2]

ii) Use your answer to part (i) to solve the equation .  

................... or ................... [2]

Write  in the form .  

Use your answer to part (a) to solve the equation   

.................... or ....................

Find the value of  and the value of .  

= ...............................................     
= ...............................................     

Write  in the form  where  and  are integers.

Hence, or otherwise, write down the coordinates of the turning point of the graph of 

Express in the form where andare integers.

is the curve with equation
The point is the maximum point on

Use your answer to part (a) to write down the coordinates of

Express  in the form  where  ,  and  are integers.

The function is such that       for

Express  in the form  where  and  are constants.

Write  in the form  where ,  and  are integers.

Express  in the form  where  and  are constants.

can be written in the form  where and are positive numbers.

Work out the values of  and .

Find the turning point of    by completing the square.  

( ...................... , .....................)