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 \(pq > \frac{1}{3} \)