The diagram shows a parallelogram - Edexcel - GCSE Maths - Question 24 - 2019 - Paper 2

To find the area of the parallelogram, use the formula: $Area = base \times height$.

Here, the base can be taken as $(2x - 1) \, cm$, and the height can be calculated using: $height = (10 - x) \, cm \times \sin(150^{\circ})$. Since $\sin(150^{\circ}) = \frac{1}{2}$, we can express the height as: $height = (10 - x) \times \frac{1}{2} = 5 - \frac{x}{2}$.

Thus, the area becomes: $Area = (2x - 1) \times (5 - \frac{x}{2})$.

Expanding this gives: $Area = (2x - 1)(5 - \frac{x}{2}) = 10x - x^{2} - 5 + \frac{x}{2} = -x^{2} + \frac{21x}{2} - 5$.

To find the condition for the area being greater than $15 \, cm^{2}$, equate: $10x - x^{2} - 5 > 15$ $-x^{2} + 10x - 20 > 0$ Multiply through by -1 (reversing the inequality): $x^{2} - 10x + 20 < 0$

Solving the quadratic equation using the discriminant: $D = b^{2} - 4ac = (-10)^{2} - 4(1)(20) = 100 - 80 = 20$ The roots are given by: $x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{20}}{2} = 5 \pm \sqrt{5}$. Hence, we have: $2x^{2} - 21x + 40 < 0$.