The curve C has equation $y = x(5 - x)$ and the line L has equation $2y = 5x + 4$ - Edexcel - A-Level Maths Pure - Question 7 - 2012 - Paper 1

To find if the curve C intersects the line L, we first express both equations in a standard form.

Starting with the equation of the curve:
$y = x(5 - x) = 5x - x^2$
Now, rewriting the line equation $2y = 5x + 4$ gives us:
$y = \frac{5}{2}x + 2$

Next, we set these equations equal to each other to find the points of intersection: $5x - x^2 = \frac{5}{2}x + 2$

Rearranging this leads to: $-x^2 + 5x - \frac{5}{2}x - 2 = 0$

To eliminate the fraction, we multiply through by 2:
$-2x^2 + 10x - 5x - 4 = 0$ $-2x^2 + 5x - 4 = 0$

Now we apply the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Where $a = -2$, $b = 5$, and $c = -4$.
Calculating the discriminant: $D = b^2 - 4ac = 5^2 - 4(-2)(-4) = 25 - 32 = -7$

Since the discriminant is negative, there are no real roots, indicating that the curves do not intersect.