1What could the values of \( x \) be in \( \sqrt{x} - 2 = 3 \), \( \sqrt{2x - 1} = 3 \) and \( \sqrt{3x + 10} - x = 0 \)?
2How can equations involving surds be solved?
3Is there a fixed rule for solving equations involving surds?
4Solve and check: \( \sqrt{2x} = 6 \)
5Solve and check: \( \sqrt{3x - 5} = 1 \)
6Solve and check: \( 6 - \sqrt{3x + 4} = 1 \)
7Solve and check: \( \sqrt[3]{5x - 3} = 3 \)
8Solve and check: \( \sqrt{3x + 6} - 3\sqrt{x - 4} = 0 \)
9Solve and check: \( \sqrt{11x^2 + 45} = 4x \)
10Solve and check: \( \sqrt{x + 7} + \sqrt{x} = 7 \)
11Solve and check: \( \sqrt{x + 5} = 5 - \sqrt{x} \)
12Solve and check: \( \sqrt{3x - 2} - \sqrt{x - 2} = 2 \)
13Solve and check: \( \sqrt{3x + 10} - x = 0 \)
14Solve and check: \( \sqrt{3x - 5} = x - 1 \)
15Solve and check: \( \sqrt{2x^2 - 7} = x + 3 \)
16Solve and check: \( \sqrt{x - 1} + \sqrt{x} = \frac{2}{\sqrt{x}} \)
17Solve and check: \( \frac{x - 1}{\sqrt{x} + 1} = 3 + \frac{\sqrt{x} + 1}{2} \)
18Solve and check: \( \sqrt{x^2 - 3x + 5} - \sqrt{x^2 - 4x + 4} = 1 \)
19Solve and check: \( \frac{\sqrt{x + 4} + \sqrt{x - 4}}{\sqrt{x + 4} - \sqrt{x - 4}} = 2 \)
20Solve and check: \( \frac{1}{\sqrt{x + 2}} - \frac{1}{\sqrt{4x + 8}} = 2 \)
21Solve and check: \( \sqrt{x^2 - 3x + 3} + \sqrt{x^2 - x + 1} = 2 \)
22Solve and check: \( \sqrt{x^2 - 2x - 4} - \sqrt{x^2 - 3x - 3} = 1 \)
23Prove that solving \( \frac{x - 1}{\sqrt{x} + 1} = 4 + \frac{\sqrt{x} - 1}{2} \) gives \( x = 81 \). When is this equation undefined? Give the reason.
24Make five other questions like these, solve them, and present them in class.
25Is \( \sqrt{49} \) a surd? Is \( \sqrt{50} \)? Give reasons, and write the surd as a mixed surd.
26Why can squaring both sides of an equation produce a false root? Give an example.
27Show that \( \sqrt{x} = -3 \) has no solution.
28Where are surds used, and why are they useful?