Given that $x > 0$ and $x \neq 25$, fully simplify $$\frac{10 + 5x - 2x^{\frac{3}{2}} - x^2}{5 - \sqrt{x}}$$ Fully justify your answer. - AQA - A-Level Maths Pure - Question 6 - 2021 - Paper 3

To simplify the numerator, we can rewrite it as:

$10 + 5x - (2x^{\frac{3}{2}} + x^2)$

Recognizing that $x^2 = \sqrt{x}^2$, we factor:

$10 + 5\sqrt{x} - \left(\sqrt{x}^2 (2\sqrt{x} + 1)\right)$

We notice that the terms have a common factor of $\sqrt{x}$ in the quadratic expression:

$\Rightarrow 10 + 5\sqrt{x} - \sqrt{x}(2\sqrt{x} + 1)$

Thus, we can group:

$= (10 + 5\sqrt{x}) - \sqrt{x}(2\sqrt{x} + 1)$

The expression can now be simplified further by cancelling the common factor of $(5 - \sqrt{x})$:

$\frac{(10 + 5\sqrt{x})}{(5 - \sqrt{x})}$

Thus, our expression simplifies to:

$2 + \sqrt{x}$