1. 1
   An engineer models the cross-sectional stress in a beam using quadratic expressions written in completed-square form. The first expression is $x^2+6x+2$. A second beam uses $2x^2+8x-3$. A third uses the completed-square form to find the vertex of a parabola $y=x^2-4x+9$.

   (a)

   Write $x^2+6x+2$ in the form $(x+a)^2+b$.

   [1 mark]

   • A$(x+3)^2-7$
   • B$(x+3)^2+2$
   • C$(x+6)^2-34$
   • D$(x+3)^2-11$

   (b)

   Write $2x^2+8x-3$ in the form $a(x+b)^2+c$.

   [1 mark]

   • A$2(x+4)^2-11$
   • B$2(x+2)^2-3$
   • C$2(x+2)^2-11$
   • D$2(x+2)^2+5$

   (c)

   By writing $y=x^2-4x+9$ in completed-square form, find the coordinates of the vertex of the parabola.

   [1 mark]

   • A$(4,9)$
   • B$(-2,5)$
   • C$(2,9)$
   • D$(2,5)$

   Total for question 1: 3 marks

2. 2
   A ball's height above the ground, in metres, is modelled by $h=x^2-8x+18$, where $x$ is horizontal distance in metres.

   (a)

   Complete the square for this expression, writing it in the form $(x+a)^2+b$.

   [2 marks]

   (b)

   Using your completed-square form, state the minimum height of the ball and the horizontal distance at which this minimum occurs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A satellite dish's cross-sectional shape is modelled by $y=3x^2-12x+7$, where $x$ and $y$ are measured in metres from a reference point.

   (a)

   A satellite dish's cross-section is modelled by $y=3x^2-12x+7$. Complete the square for this expression, writing it in the form $a(x+b)^2+c$, showing full working.

   [3 marks]

   (b)

   Use your completed-square form to solve $3x^2-12x+7=0$, giving your answers to 2 decimal places.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   An architect models the arch of a bridge with the quadratic $y=-2x^2+20x-32$, where $y$ is the height above the road in metres and $x$ is the horizontal distance in metres. The architect needs to find the maximum height of the arch and the width of the arch at road level.

   (a)

   Rewrite $y$ in completed-square form, showing every step, by first factoring out $-2$.

   [4 marks]

   (b)

   Using your completed-square form, find the maximum height of the arch and the horizontal distance at which it occurs.

   [4 marks]

   (c)

   Find the width of the arch at road level (where $y=0$) by solving $-2(x-5)^2+18=0$, showing all steps and interpreting your answer in context.

   [5 marks]

   Total for question 4: 13 marks

End of questions