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 - Edexcel - A-Level Maths Pure - Question 10 - 2016 - Paper 1

To prove that the quadratic inequality $4p^2 - 20p + 9 < 0$ holds true for the values of $p$ ensuring that the line does not intersect the curve, we start by analyzing the discriminant:

The discriminant, $D$, of a quadratic equation in the form $ax^2 + bx + c$ is given by: $D = b^2 - 4ac$ For our inequality:

• $a = 4$
• $b = -20$
• $c = 9$

Now, calculating the discriminant: $D = (-20)^2 - 4(4)(9)$ $D = 400 - 144$ $D = 256$

Since the discriminant is positive, the quadratic can change values, and we will find the roots: $p = \frac{-b \pm \sqrt{D}}{2a} = \frac{20 \pm \sqrt{256}}{2 \cdot 4}$ $p = \frac{20 \pm 16}{8}$

Calculating the roots:

1. $p = \frac{36}{8} = 4.5$
2. $p = \frac{4}{8} = 0.5$

To determine where the quadratic is negative, we observe the quadratic opens upwards (since $a = 4 > 0$). Thus, it is negative between its roots: $0.5 < p < 4.5$