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Let's work through each question in sequence.

### Question 7
We are given:
\[
\frac{1}{\sqrt{x} + 2} - \frac{1}{\sqrt{4x} + 8} = \frac{a}{b(\sqrt{x} + 2)}
\]
and need to find values of \( a \) and \( b \), then solve:
\[
\frac{1}{\sqrt{x} + 2} - \frac{1}{\sqrt{4x} + 8} = 2.
\]

1. Start by simplifying \(\frac{1}{\sqrt{x} + 2} - \frac{1}{\sqrt{4x} + 8}\) by making a common denominator.
2. Compare terms to identify \( a \) and \( b \).
3. Substitute the values of \( a \) and \( b \) into the equation and solve for \( x \).

### Question 8
We are asked to solve:
\[
x = \sqrt{4x + 5}.
\]

1. Square both sides to eliminate the square root.
2. Rearrange the resulting quadratic equation to standard form.
3. Solve for \( x \) using the quadratic formula or by factoring if possible.

### Question 9
For this question:
If \( x = \sqrt{a} + \frac{1}{\sqrt{a}} + 1 \), express \( x^2 - 2x \) in terms of \( a \).

1. Start by expanding \( x^2 \).
2. Substitute and simplify using the definition of \( x \) in terms of \( a \).
3. Collect terms to express \( x^2 - 2x \) in terms of \( a \).

### Question 10
Given:
\[
(a + \sqrt{3})(b - \sqrt{3}) = 7 + 3\sqrt{3},
\]
find positive integer values for \( a \) and \( b \).

1. Expand the left side by multiplying terms.
2. Separate real and irrational components.
3. Match terms to identify values of \( a \) and \( b \).

### Question 11
The length, width, and height of an open rectangular box are expressed in terms of \( x \) (the height). The box dimensions are:
- Length: \( x + 2 \)
- Width: \( x - 2 \)
- Height: \( x \)

(i) **Diagonal [IC]:**
1. Use the Pythagorean theorem within the base rectangle to find the diagonal \( [IC] \), which is the diagonal across the base with sides \( x + 2 \) and \( x - 2 \).

(ii) **Diagonal [ID]:**
1. Use the Pythagorean theorem in three dimensions with sides \( x \), \( x + 2 \), and \( x - 2 \) to find \( [ID] \), the diagonal from \( I \) to \( D \).
2. Given \( |ID| = \sqrt{56} \), set up an equation to solve for \( x \).

Would you like a detailed solution to any specific question from these steps?

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#### Suggested Further Questions:
1. What are the properties of surds that help in simplifying equations like in question 7?
2. How does squaring both sides of an equation (like in question 8) affect possible solutions?
3. How can terms involving square roots be eliminated or simplified when solving expressions in terms of a single variable, as in question 9?
4. What strategies help when matching terms involving surds, as in question 10?
5. How can the Pythagorean theorem be applied in three-dimensional geometry problems like question 11?

#### Tip:
For solving equations involving square roots, always consider squaring as a strategy, but remember to verify solutions as squaring can introduce extraneous solutions.

Solve questions 7 to 11 related to algebra and geometry, including solving equations with square roots and finding diagonals in a rectangular box.

Explore solutions to complex algebraic and geometric problems, including simplifying surds, solving quadratic equations, and finding 3D diagonals in a rectangular box. Learn step-by-step methods for high school math challenges, suitable for students in Grades 10-12.

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