Cambridge IGCSE Additional Mathematics · Coursebook Exercise 2.2, Q1–16
Math Exercise 2.2 — Completing the Square
Graded 2026-06-18 · Sys4Ethan (Claude Opus 4.8 vision, human-verified)
100%
✅ Correct 16
◐ Partial 0
✗ Needs fixing 0
○ Not attempted 0
Total 16
QQ1 (sketch via symmetry, a–f)
✓ Correct
roots + axis of symmetry
Factorised (or used the formula for e) and sketched all six, with correct axis crossings — incl. the $-2\pm\sqrt3$ and downward $15+2x-x^2$ cases.
QQ2 (complete the square, a–h)
✓ Correct
$x^2+bx \to (x-m)^2+n$
All eight correct.
QQ3 (complete the square +c, a–h)
✓ Correct
$(x-m)^2+n$
All eight correct, incl. $-\tfrac94+4=\tfrac74$ type fraction work.
QQ4 (leading coeff ≠ 1, a–h)
✓ Correct
$a(x-p)^2+q$
All eight correct — e.g. $2x^2+7x-3=2(x+\tfrac74)^2-\tfrac{73}{8}$, $3x^2-x+6=3(x-\tfrac16)^2+\tfrac{71}{12}$.
QQ5 ($m-(x-n)^2$, a–d)
✓ Correct
negative quadratics
All four correct.
QQ6 ($a-(x+b)^2$, a–d)
✓ Correct
negative quadratics
All four correct.
QQ7 ($a-p(x+q)^2$, a–d)
✓ Correct
negative, coeff ≠ 1
All four — incl. $2+5x-3x^2=\tfrac{49}{12}-3(x-\tfrac56)^2$.
QQ8 ($4x^2+2x+5$, meets x-axis?)
✓ Correct
completed square → roots
$4(x+\tfrac14)^2+\tfrac{19}{4}$; min value $\tfrac{19}{4}>0$, so it does not meet the x-axis ✓.
QQ9 (stationary point)
✓ Correct
vertex from completed square
$2(x-2)^2-7$, stationary point $(2,-7)$ ✓.
QQ10 (min value + domain)
✓ Correct
min + domain for inverse
Min $-\tfrac{21}{4}$ at $x=\tfrac12$ ✓.
QQ11 (range)
✓ Correct
range of a downward quadratic
$\tfrac{89}{8}-2(x+\tfrac74)^2$, range $f(x)\le\tfrac{89}{8}$ ✓.
QQ12 (stationary point + sketch)
✓ Correct
completed square → vertex
$\tfrac{37}{2}-2(x-\tfrac32)^2$, stationary point $(\tfrac32,\tfrac{37}{2})$ ✓. (Make sure the sketch opens downward — it's a maximum.)
QQ13 (range + inverse domain)
✓ Correct
range + domain
Max point, range and the $0\le x\le7$ domain all handled ✓.
QQ14 (domain for inverse)
✓ Correct
domain for one-to-one
$2(x-2)^2-5$; $x\ge2$ (or $x\le2$) ✓.
QQ15 (smallest m)
✓ Correct
vertex → smallest domain
$m=-\tfrac34$ ✓.
QQ16 (inverse function)
✓ Correct
find $f^{-1}(x)$
$5-(x-2)^2$, $(2,5)$ max, and $f^{-1}(x)=2+\sqrt{5-x}$ ✓ — excellent.
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