Solutions — Parabolas 7G Worksheet
Year 10 Mathematics Core · Cambridge Ch 7, §7G · Mr Wong
ANSWER KEY
Warm-up
Q1. Vertical line meets parabola:
Q2. Rearranged:
a $x^2+3x+6=0$
b $x^2-5x+3=0$
Part A — Solving by substitution
Q3. Points of intersection:
a $(0,0)$ and $(3,9)$
b $(-1,-1)$ and $(2,2)$
c $(1,2)$ only (tangent)
Q4. $x^2=2x-1 \Rightarrow x^2-2x+1=0 \Rightarrow (x-1)^2=0$, so $x=1$, $y=1$.
· One point $(1,1)$ — the line is a tangent.
Part B — Working & sketching
Q5. $x^2-x-2=x+1$:
a $x^2-2x-3=0$
b $x=-1$ or $x=3$
c $(-1,0)$ and $(3,4)$
Part C — Using the quadratic formula
Q6. $x^2+2x-1=x+3$:
a $x^2+x-4=0$
b $(1.56,4.56)$ and $(-2.56,0.44)$
Part D — Discriminant
Q7. Number of intersection points:
a $\Delta=0$ → one (tangent)
b $\Delta=-15$ → none
c $\Delta=-3$ → none
d $\Delta=-24$ → none
Challenge
Q8. $x^2=x+k \Rightarrow x^2-x-k=0$. Tangent means $\Delta=0$: $(-1)^2-4(1)(-k)=1+4k=0$.