Solving quadratics and completing the squareEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f(x)=x^2-10x+7$.

   (a)

   Which of the following is $f(x)$ in the form $(x+p)^2+q$?

   [1 mark]

   • A$(x-5)^2+32$
   • B$(x-5)^2-18$
   • C$(x+5)^2-18$
   • D$(x-10)^2-93$

   (b)

   What are the solutions of $f(x)=0$?

   [1 mark]

   • A$x=5\pm4\sqrt2$
   • B$x=-5\pm3\sqrt2$
   • C$x=5\pm18$
   • D$x=5\pm3\sqrt2$

   (c)

   Hence state the minimum value of $f(x)$ and the value of $x$ at which it occurs.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g(x)=2x^2+12x+5$.

   (a)

   Which of the following is $g(x)$ in the form $a(x+p)^2+q$?

   [1 mark]

   • A$2(x+3)^2-13$
   • B$2(x+6)^2-67$
   • C$2(x+3)^2+23$
   • D$2(x-3)^2-13$

   (b)

   What are the coordinates of the minimum point of $y=g(x)$?

   [1 mark]

   • A$(3,-13)$
   • B$(-3,13)$
   • C$(-3,-13)$
   • D$(-6,-67)$

   (c)

   Solve $g(x)=0$, giving your answers in exact form.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f(x)=3x^2-7x-6$.

   (a)

   Solve $f(x)=0$ by factorising.

   [3 marks]

   (b)

   Solve $f(x)=5$, giving your answers to $3$ significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rectangular field has width $w$ metres and length $(2w+5)$ metres. The area of the field is $1375$ m$^2$.

   (a)

   (i) Show that $2w^2+5w-1375=0$.
   (ii) Solve this equation to find the width of the field.
   (iii) Find the perimeter of the field.

   [6 marks]

   (b)

   The owner builds a path of uniform width $t$ metres around the outside of the field, so that the field and path together have an area of $2750$ m$^2$. Using your answer to (a), form an equation in $t$ and solve it by completing the square, giving $t$ to $3$ significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions