The straight line with equation $y = 3x - 7$ does not cross or touch the curve with equation $y = 2p x^2 - 6px + 4p$, where $p$ is a constant - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 1

To demonstrate that the quadratic $4p^2 - 20p + 9 < 0$, we will use the discriminant method.

The general form of a quadratic is given by: $ax^2 + bx + c$ where $a = 4$, $b = -20$, and $c = 9$.

First, we calculate the discriminant $D$ using the formula: $D = b^2 - 4ac$ Substituting the values: $D = (-20)^2 - 4(4)(9)$ $D = 400 - 144$ $D = 256$

Since the discriminant is positive ($D > 0$), the quadratic has two distinct real roots. Next, we find the roots using the quadratic formula: $p = \frac{-b \pm \sqrt{D}}{2a}$ Substituting the values we have: $p = \frac{20 \pm \sqrt{256}}{8}$ $p = \frac{20 \pm 16}{8}$ Thus, we obtain: $p_1 = \frac{36}{8} = 4.5$ $p_2 = \frac{4}{8} = 0.5$

Now, analyzing the roots, the quadratic is less than zero between these two values: $0.5 < p < 4.5$ Therefore, we have shown that $4p^2 - 20p + 9 < 0$ for $p$ in the range $0.5 < p < 4.5$.