Solutions — 5I Worksheet
Year 10 Mathematics Core · Cambridge Ch 5, §5I · Mr Wong
ANSWER KEY
Warm-up — completing the square
Q1. Add $\left(\tfrac b2\right)^2$:
a $9$
b $25$
c $49$
d $\tfrac94$
Part A — Completed-square form
Q2.
a $(x-1)^2-5$
b $(x+4)^2-13$
Part B — Solving (whole-number $b$)
Q3.
a $x=1\pm\sqrt5$ $(x-1)^2=5$
b $x=-4\pm\sqrt{13}$ $(x+4)^2=13$
c $x=3\pm\sqrt5$ $(x-3)^2=5$
d $x=-1\pm\sqrt2$ $(x+1)^2=2$
Part C — Fractional middle term
Q4. $x^2-x-1=0$:
· $\left(x-\tfrac12\right)^2=\tfrac54$, so
$x=\dfrac{1\pm\sqrt5}{2}$
Q5. $x^2-5x+2=0$:
· $\left(x-\tfrac52\right)^2=\tfrac{17}{4}$, so
$x=\dfrac{5\pm\sqrt{17}}{2}$
Part D — How many solutions?
Q6.
a no real solution $(x-1)^2=-7$
b no real solution $x^2=-5$
Challenge (Extension, $a\ne 1$)
Q7. $3x^2-12x=15 \Rightarrow x^2-4x-5=0$:
· $(x-2)^2-9=0 \Rightarrow (x-2)^2=9 \Rightarrow x-2=\pm 3$, so
$x=5$ or $x=-1$ (these factorise neatly).