The straight line with equation y = 3x – 7 does not cross or touch the curve with equation y = 2px^2 – 6px + 4p, where p is a constant. Show that 4p^2 – 20p + 9 < 0.

The main piece of information this question gives us is that the two lines do not cross or touch. From this we can immediately see that we will need to use the discriminant of the quadratic formula b² - 4ac. To start with treat the curves as simultaneous equations and bring all terms to one side, 2px²-6px-3x+4p+7=0. Now group the terms together to form a quadratic equation you will recognise. E.g. x²(2p)+x(-6p-3)+(4p+7)=0. As the lines don't cross we know there will be no real roots to this equation so b²-4ac < 0.

By plugging the constants into this equation we get (-6p-3)²-4(2p)(4p+7)< 0 and this simplifies to 4p^2  – 20p + 9 < 0 as required.