Let’s have a look at another question from Methodist Girls School
(MGS Prelim 2016 AMath Paper 2 Q 1b)
This question is in line with the 4047 Syllabus where students have to know
“Conditions for \(ax^2+bx+c\) to be always positive (or always negative)”
Above extracted from SEAB.
Question: Determine the conditions for \(p\) and \(q\) such that the curve \(y=px^2-2x+3q\) lies entirely above the\(x\)axis, where \(p\) and \(q\) are constants. [3 marks]
Solution: The keyword is “entirely above”, implying that the graph is above the axis. We expect a Happy Face that is not intersecting the x-axis.
\(\Rightarrow\) Happy Face is ABOVE axis
\(\Rightarrow\) No Real Solutions
\(\Rightarrow\) \(b^2-4ac < 0\)
Observe that \(a=p\), \(b=-2\) and \(c=3q\),
\(b^2-4ac < 0 \)
\((-2)^2 – 4(p)(3q) < 0 \)
\(4-12pq<0\)
\(1-3pq<0\)
\(\frac{1}{3} < pq \)
Since the graph is a happy face, we know that the coefficient of \(x^2\) is positive and \(p>0\).
Therefore, \( p>0\) and \(pq > \frac{1}{3} \)