Show that the discriminant of $f(x)$ is $100 - 4{p^2}$.

Find the values of $p$ so that $f(x) = 0$ has two equal roots.

correct substitution into ${b^2} - 4ac$     A1

eg$\;\;\;{(10 - p)^2} - 4(p)\left( {\frac{5}{4}p - 5} \right)$

correct expansion of each term     A1A1

eg$\;\;\;100 - 20p + {p^2} - 5{p^2} + 20p,{\text{ }}100 - 20p + {p^2} - (5{p^2} - 20p)$

$100 - 4{p^2}$     AG     N0

[3 marks]

recognizing discriminant is zero for equal roots     (R1)

eg$\;\;\;D = 0,{\text{ }}4{p^2} = 100$

correct working     (A1)

eg$\;\;\;{p^2} = 25$, $1$ correct value of $p$

both correct values $p =  \pm 5$     A1     N2

[3 marks]

Total [6 marks]

Many candidates were able to identify the discriminant correctly and continued with good algebraic manipulation. A commonly seen mistake was identifying the constant as $\frac{5}{4}p$ instead of $\frac{5}{4}p - 5$. Mostly a correct approach to part b) was seen $(\Delta  = 0)$, with the common error being only one answer given for $p$, even though the question said values (plural).

Many candidates were able to identify the discriminant correctly and continued with good algebraic manipulation. A commonly seen mistake was identifying the constant as $\frac{5}{4}p$ instead of $\frac{5}{4}p - 5$. Mostly a correct approach to part b) was seen $(\Delta  = 0)$, with the common error being only one answer given for $p$, even though the question said values (plural).