Quadratic Equations (Cambridge (CIE) O Level Maths): Exam Questions

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]

Solve by factorisation $10r^2-23r+9=0$.    

$r$ = ................... or $r$ = ................... 

$x^2-12x+a={(x+b)}^2$  
Find the value of $a$ and the value of $b$.  

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

Write $x^2–6x–19$ in the form ${(x–a)}^2+b.$

Using your answer to part (a), write down the coordinates of the turning point of the graph of $y=x^2–6x–19$ .

Write $x^2-10x+23$ in the form ${\left(x-a\right)}^2-b$.

Using your answer to part (a), write down the coordinates of the turning point of the graph of $y=x^2-10x+23$.

The solutions of the equation $x^2+bx+c=0$ are $\frac{-7+\sqrt{61}}{2}$ and $\frac{-7-\sqrt{61}}{2}.$  
Find the value of $b$ and the value of $c$.  

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

  The difference between the areas of the two rectangles is 62 cm².

i) Show that $x^2+2x-63=0.$  

[3]

ii) Factorise $x^2+2x-63.$  

[2]

iii) Solve the equation $x^2+2x-63=0$ to find the difference between the perimeters of the two rectangles.  

.............................................. cm [2]

The lengths of the sides are in centimetres.
The area of triangle T₁ is equal to the area of triangle T₂.
Work out the value of x, giving your answer in the form $a+\sqrt{b}$ where a and b are integers.