Completing the Square (Edexcel IGCSE Maths A: Higher): Exam Questions

The expression $x^2—8x+21$ can be written in the form ${\left(x-a\right)}^2+b$ for all values of $x$.

Find the value of $a$ and the value of $b$.

The equation of a curve is $y=\mathrm{f}\left(x\right)$ where $\mathrm{f}\left(x\right)=x^2—8x+21$ The diagram shows part of a sketch of the graph of $y=\mathrm{f}\left(x\right)$.

The minimum point of the curve is $M$.

Write down the coordinates of $M$.

Given that $x^2-6x+1={\left(x-a\right)}^2-b$ for all values of $x$.

(i) Find the value of $a$ and the value of $b$.

[2]

(ii) Hence write down the coordinates of the turning point of the graph of $y=x^2-6x+1$.

[1]

Given that $a,b$ and $c$ are integers,

express  $3x^2+12x+19$ in the form $a{(x+b)}^2+c$

Express $4x^2–8x+7$  in the form  $a{(x+b)}^2+c$ where $a,b$ and $c$ are integers.

Write $3x^2-12x+7$ in the form $a{(x+b)}^2+c$

The line $\mathbf{L}$ is the line of symmetry of the curve with equation $y=3x^2-12x+7$

Using your answer to part (a) or otherwise, write down an equation of $\mathbf{L}$.

Express $x^2+6\sqrt{2}x–1$ in the form ${(x+a)}^2+b$

Show your working clearly.

The equation of a curve is$y=x^2+14x+52$

By completing the square, work out the coordinates of the turning point.
You must show your working.

(i) Write $x^2+4x-16$ in the form ${\left(x+a\right)}^2-b$.

[3]

(ii) Solve the equation $x^2+4x-16=0$.
Give your answers in surd form as simply as possible.

$x$ = ........... or $x$ = .......... [4]

$x^2-12x+a={(x+b)}^2$  

Find the value of $a$ and the value of $b$.  

         $a$ = ............................................... 

$b$ = ...............................................

(i) Write $x^2+8x-9$ in the form ${\left(x+k\right)}^2+h$.  

[2]

(ii) Use your answer to part (i) to solve the equation $x^2+8x-9=0$.  

$x=$................... or $x=$................... [2]

Write $x^2-18x-27$ in the form ${\left(x+k\right)}^2+h$.

Use your answer to part (a) to solve the equation $x^2-18x-27=0.$  

$x=$.................... or $x=$.................... 

The curve C has equation   $y=x^2-8x+10$

Using the axes below, sketch the curve C.

On your sketch show clearly

(i) the exact coordinates of any points of intersection of C with the coordinate axes,

(ii) the coordinates of the turning point.

Write  $x^2+10x+14$  in the form  ${\left(x+a\right)}^2+b.$

Sketch the graph of  $y=x^2+10x+14$,  indicating the coordinates of the turning point.

Express $3x^2–6x+5$ in the form $a{(x–b)}^2+c$

Write $2x^2+16x+35$ in the form $a{\left(x+b\right)}^2+c$ where $a,b,$ and $c$ are integers.

Hence, or otherwise, write down the coordinates of the turning point of the graph of $y=2x^2+16x+35$

Express $7+12x–3x^2$ in the form $a+b{(x+c)}^2$ where $a,b$ and$c$are integers.

$\mathbf{C}$ is the curve with equation$y=7+12x–3x^2$

The point $A$ is the maximum point on $\mathbf{C}$

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

Express $7-12x-2x^2$ in the form $a+b{\left(x+c\right)}^2$ where $a$ , $b$ and $c$ are integers.

The function$\mathrm{f}$ is such that $\mathrm{f}\left(x\right)=5+6x–x^2$      for $x⩽3$

Express $5+6x–x^2$ in the form $p–{(x–q)}^2$ where $p$ and $q$ are constants.

Write $5+12x-2x^2$ in the form $a+b{(x+c)}^2$ where $a$ , $b$ and $c$ are integers.

Express $7–4x–x^2$ in the form $p–{(x+q)}^2$ where $p$ and $q$ are constants.

$2x^2-6x+5$ can be written in the form $a{(x-b)}^2+c$

where $a,b$ and$c$ are positive numbers. Work out the values of  $a,b$ and $c$.

$\begin{array}{rcl}a& =& ...................\\ b& =& ...................\\ c& =& ...................\end{array}$

Find the turning point of  $y=x^2+4x-3$  by completing the square.  

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