SURDS

 Below is a comprehensive, structured note on surds. This note covers their definition and types, the fundamental laws and operations (addition, subtraction, multiplication, division), techniques for simplification and rationalization (using conjugates), and finally 10 worked examples modeled on JAMB exam questions.

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1. Introduction to Surds

Definition:
A surd is an irrational number expressed in root form that cannot be simplified to a rational number. For example, since √2, √3, and √5 do not simplify to whole numbers or exact fractions, they are surds. In contrast, √4 equals 2 and is not considered a surd.

Types of Surds:

• Pure Surds: Contain only an irrational part (e.g. √7, ∛2).
• Mixed Surds: Have both a rational coefficient and an irrational root (e.g. 3√5, 2√2).
• Compound Surds: Are expressions formed by the sum or difference of surds (e.g. √3 + √2).

(For a historical perspective and further details on surd types, see citeturn0search10.)

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2. Laws of Surds

The basic rules governing surds include:

1. Multiplication Law:

   $$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$$

   Example: √2 × √3 = √6.

2. Division Law:

   $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$

   (provided $b \ne 0$).

3. Exponential Form:
   A square root can be written as an exponent of $1/2$:

   $$\sqrt{a} = a^{\frac{1}{2}} \quad \text{and generally} \quad \sqrt[n]{a} = a^{\frac{1}{n}}.$$

4. Addition/Subtraction:
   Surds can be added or subtracted only if they are like surds (i.e. have the same radicand). For example:

   $$2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}, \quad \text{but} \quad \sqrt{2} + \sqrt{3} \quad \text{cannot be combined.}$$

(These laws are standard; see citeturn0search8 for a further explanation.)

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3. Operations on Surds

3.1 Addition and Subtraction

• Like Surds:
  Only surds with the same radicand can be added or subtracted.
  Example: $$4\sqrt{2} - 3\sqrt{2} = \sqrt{2}.$$

3.2 Multiplication

• Multiply the coefficients and the surd parts separately:
  Example: $$2\sqrt{3} \times 4\sqrt{5} = (2 \times 4)\sqrt{3 \times 5} = 8\sqrt{15}.$$

3.3 Division

• Divide the coefficients and write the surd as a quotient under a single radical:
  Example: $$\frac{6\sqrt{8}}{3\sqrt{2}} = 2 \times \frac{\sqrt{8}}{\sqrt{2}} = 2\sqrt{\frac{8}{2}} = 2\sqrt{4} = 2 \times 2 = 4.$$

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4. Simplification of Surds (5 Examples)

1. Simplify $\sqrt{18}$:

   $$\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9}\sqrt{2} = 3\sqrt{2}.$$

2. Simplify $\sqrt{50}$:

   $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}.$$

3. Simplify $\sqrt{72}$:

   $$\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}.$$

4. Simplify $\sqrt{98}$:

   $$\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}.$$

5. Simplify $\sqrt{32}$:

   $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}.$$

(These techniques follow the method of extracting the largest perfect square factor; see citeturn0search16 for more details.)

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5. Rationalisation of Surds

Rationalisation is the process of eliminating surds from the denominator of a fraction.

• Single-term Denominator:
  Multiply numerator and denominator by the surd in the denominator.
  Example: $$\frac{8}{\sqrt{3}} = \frac{8\sqrt{3}}{3}.$$
• Binomial Denominator:
  When the denominator is of the form $a + b\sqrt{c}$, multiply by its conjugate $a - b\sqrt{c}$.
  Example: $$\frac{5}{2+\sqrt{3}} = \frac{5(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} = \frac{5(2-\sqrt{3})}{4-3} = 5(2-\sqrt{3}).$$

(For a deeper discussion on rationalisation using conjugates, see citeturn0search4.)

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6. Conjugate Surds

Definition:
The conjugate of a binomial surd $a + b\sqrt{c}$ is $a - b\sqrt{c}$. Multiplying conjugate pairs results in a rational number:

$$(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c.$$

Example:
For $3+\sqrt{2}$, the conjugate is $3-\sqrt{2}$ and

$$(3+\sqrt{2})(3-\sqrt{2}) = 9 - 2 = 7.$$

(See citeturn0search2 for an explanation of conjugates in the context of rationalisation.)

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7. 10 JAMB Exam Worked Examples

Below are ten worked examples—each with detailed steps—to help you master surd problems as seen in JAMB exams.

Example 1: Simplify $\sqrt{18} + \sqrt{8}$

1. Simplify each surd:
   • $\sqrt{18} = 3\sqrt{2}$
   • $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
2. Add like surds: $$3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}.$$

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Example 2: Multiply $2\sqrt{3} \times 4\sqrt{5}$

1. Multiply coefficients: $2 \times 4 = 8$.
2. Multiply surds: $\sqrt{3} \times \sqrt{5} = \sqrt{15}$.
3. Answer: $8\sqrt{15}$.

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Example 3: Divide $\frac{\sqrt{50}}{\sqrt{2}}$

1. Simplify numerator: $\sqrt{50} = 5\sqrt{2}$.
2. Divide: $$\frac{5\sqrt{2}}{\sqrt{2}} = 5.$$

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Example 4: Rationalise $\frac{5}{2+\sqrt{3}}$

1. Multiply numerator and denominator by the conjugate $2-\sqrt{3}$: $$\frac{5}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} = \frac{5(2-\sqrt{3})}{(2)^2-(\sqrt{3})^2} = \frac{5(2-\sqrt{3})}{4-3} = 5(2-\sqrt{3}).$$
2. Answer: $10 - 5\sqrt{3}$.

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Example 5: Simplify $\sqrt{72} - \sqrt{50}$

1. Simplify:
   • $\sqrt{72} = 6\sqrt{2}$
   • $\sqrt{50} = 5\sqrt{2}$
2. Subtract: $$6\sqrt{2} - 5\sqrt{2} = \sqrt{2}.$$

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Example 6: Expand $(\sqrt{3} + \sqrt{2})^2$

1. Use the binomial square formula: $$(\sqrt{3} + \sqrt{2})^2 = (\sqrt{3})^2 + 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2.$$
2. Calculate: $$= 3 + 2\sqrt{6} + 2 = 5 + 2\sqrt{6}.$$

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Example 7: Solve for $k$ in $\sqrt{28} + \sqrt{112} - \sqrt{k} = \sqrt{175}$

1. Express each surd in terms of $\sqrt{7}$:
   • $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$
   • $\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$
   • $\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}$
2. Substitute: $$2\sqrt{7} + 4\sqrt{7} - \sqrt{k} = 5\sqrt{7} \quad \Rightarrow \quad 6\sqrt{7} - \sqrt{k} = 5\sqrt{7}.$$
3. Solve: $$\sqrt{k} = 6\sqrt{7} - 5\sqrt{7} = \sqrt{7} \quad \Rightarrow \quad k = 7.$$

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Example 8: Simplify $\frac{8\sqrt{18}}{3\sqrt{2}}$

1. Simplify $\sqrt{18}$: $$\sqrt{18} = 3\sqrt{2}.$$
2. Substitute: $$\frac{8 \times 3\sqrt{2}}{3\sqrt{2}} = \frac{24\sqrt{2}}{3\sqrt{2}} = 8.$$

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Example 9: Express $\sqrt{50} + \sqrt{18}$ in simplest form

1. Simplify each:
   • $\sqrt{50} = 5\sqrt{2}$
   • $\sqrt{18} = 3\sqrt{2}$
2. Add: $$5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}.$$

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Example 10: Rationalise and simplify $\frac{3\sqrt{2}}{4-\sqrt{3}}$

1. Multiply numerator and denominator by the conjugate $4+\sqrt{3}$: $$\frac{3\sqrt{2}(4+\sqrt{3})}{(4-\sqrt{3})(4+\sqrt{3})} = \frac{3\sqrt{2}(4+\sqrt{3})}{4^2-(\sqrt{3})^2} = \frac{3\sqrt{2}(4+\sqrt{3})}{16-3} = \frac{3\sqrt{2}(4+\sqrt{3})}{13}.$$
2. Final answer: $$\frac{12\sqrt{2}+3\sqrt{6}}{13}.$$

(Several JAMB-style surd problems and rationalisation steps are similar to those found in resources such as SchoolNgr and JAMB tutorials.

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Conclusion

This note has reviewed the key concepts and laws of surds along with strategies for:

• Simplifying surds (by extracting perfect square factors),
• Operating on surds (addition, subtraction only for like terms; multiplication and division directly using radical laws),
• Rationalising denominators (using conjugates for binomial surds), and
• Solving exam-style problems (with 10 worked examples in a JAMB context).

With this solid foundation and practice through these examples, you should be well-prepared to tackle surd problems in exam

Below is a set of 50 multiple‐choice JAMB practice questions covering the following topics:
• Indices
• Logarithms
• Number Bases
• Percentage Error
• Quadratic Equations
• Surds

Each question has four options (A–D).

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INDICES (Questions 1–10)

1. Simplify: $2^3 \times 2^4$
    A. $2^6$  B. $2^7$  C. $2^8$  D. $2^9$

2. Simplify: $\frac{3^5}{3^2}$
    A. $3^2$  B. $3^3$  C. $3^4$  D. $3^7$

3. Evaluate: $5^{-2}$
    A. $\frac{1}{25}$  B. $-25$  C. $25$  D. $\frac{1}{5}$

4. Simplify: $(2^3)^4$
    A. $2^{12}$  B. $2^7$  C. $2^{64}$  D. $2^{1}$

5. If $a^2 = 16$ (with $a>0$), find $a^3$.
    A. 32  B. 64  C. 8  D. 16

6. Simplify: $10^3 \div 10^5$
    A. $10^2$  B. $10^{-2}$  C. $10^{-8}$  D. $10^8$

7. Simplify: $4^2 \times 4^{-3}$
    A. $4^{-1}$  B. $4^5$  C. $4$  D. $4^0$

8. Find $x$: If $2^x = 32$, then
    A. 3  B. 4  C. 5  D. 6

9. Simplify: $\frac{6^3}{6^3}$
    A. 1  B. 6  C. 0  D. $6^6$

10. Simplify: $(3^{-1})^{-2}$
     A. $3^2$  B. $3^{-2}$  C. $3^{-1}$  D. $3^3$

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LOGARITHMS (Questions 11–20)

11. Solve for $x$: $\log_2 8 = x$
     A. 2  B. 3  C. 4  D. 5

12. Evaluate: $\log_{10} 100$
     A. 1  B. 2  C. 10  D. 0

13. Express: $\log_b (MN)$ in terms of $\log_b M$ and $\log_b N$
     A. $\log_b M + \log_b N$  B. $\log_b M - \log_b N$  C. $\log_b M \times \log_b N$  D. $\frac{\log_b M}{\log_b N}$

14. Simplify: $\log_3 27$
     A. 2  B. 3  C. 4  D. 9

15. Find $x$: If $\log_5 125 = x$, then
     A. 2  B. 3  C. 4  D. 5

16. Solve: $\log_{10} x = 2$
     A. 10  B. 50  C. 100  D. 200

17. Simplify: $\log_a 1$
     A. 0  B. 1  C. $a$  D. Undefined

18. Given: $\log_2 32 = 5$. Then find $\log_2 16$
     A. 2  B. 3  C. 4  D. 5

19. Express: $\log_b \frac{1}{M}$ in terms of $\log_b M$
     A. $-\log_b M$  B. $\frac{1}{\log_b M}$  C. $\log_b M$  D. $\log_b M - 1$

20. If $\log_4 x = 3$, find $x$:
     A. 64  B. 16  C. 81  D. 256

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NUMBER BASES (Questions 21–30)

21. Convert (Binary to Decimal): 1010₂ equals
     A. 10  B. 11  C. 12  D. 13

22. Convert (Decimal to Binary): 15 in decimal equals
     A. 1110₂  B. 1111₂  C. 1101₂  D. 1010₂

23. Convert (Decimal to Hexadecimal): 26 in decimal equals
     A. 1A  B. 1B  C. 16  D. 1C

24. Convert (Octal to Decimal): 17₈ equals
     A. 13  B. 15  C. 14  D. 16

25. Convert (Binary to Decimal): 1101₂ equals
     A. 11  B. 12  C. 13  D. 14

26. Convert (Decimal to Binary): 20 in decimal equals
     A. 10100₂  B. 10110₂  C. 10010₂  D. 11000₂

27. Convert (Decimal to Hexadecimal): 31 in decimal equals
     A. 1F  B. 1E  C. 20  D. 2F

28. In base-8, the digit 7 represents
     A. 5  B. 6  C. 7  D. 8

29. Convert (Hexadecimal to Decimal): The digit A in hex equals
     A. 9  B. 10  C. 11  D. 12

30. Convert (Decimal to Base-4): 8 in decimal equals
     A. 20₄  B. 22₄  C. 10₄  D. 12₄

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PERCENTAGE ERROR (Questions 31–35)

31. If the measured value is 98 and the true value is 100, the percentage error is:
     A. 1%  B. 2%  C. 5%  D. 10%

32. A length is measured as 45 cm when the true length is 50 cm. The percentage error is:
     A. 5%  B. 8%  C. 10%  D. 12%

33. If the true value is 200 and the experimental value is 190, the percentage error is:
     A. 4%  B. 5%  C. 10%  D. 2%

34. An error of 3 units on a true value of 75 gives a percentage error of:
     A. 2%  B. 4%  C. 5%  D. 3%

35. A scale reads 48 kg when the actual weight is 50 kg. The percentage error is:
     A. 2%  B. 4%  C. 5%  D. 8%

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QUADRATIC EQUATIONS (Questions 36–45)

36. Solve: $x^2 - 5x + 6 = 0$
     A. $x = 2$ or $3$  B. $x = 1$ or $6$  C. $x = -2$ or $-3$  D. $x = -1$ or $-6$

37. Solve: $2x^2 + 3x - 2 = 0$
     A. $x = \frac{1}{2}$ or $x = -2$  B. $x = -\frac{1}{2}$ or $x = 2$  C. $x = 1$ or $x = -2$  D. $x = -1$ or $x = \frac{2}{3}$

38. Solve: $x^2 + 4x + 4 = 0$
     A. $x = -2$ (double root)  B. $x = 2$ (double root)  C. $x = -4$  D. $x = 0$

39. Find the discriminant of: $x^2 - 6x + 8 = 0$
     A. 4  B. 9  C. 16  D. 25

40. For $3x^2 - 2x - 1 = 0$, the sum of the roots is:
     A. $\frac{2}{3}$  B. $-\frac{2}{3}$  C. $\frac{1}{3}$  D. $-\frac{1}{3}$

41. The product of the roots of: $x^2 + 7x + 12 = 0$ is:
     A. 7  B. 12  C. 0  D. 1

42. Solve: $x^2 - 2x - 3 = 0$
     A. $x = -1$ or $3$  B. $x = 1$ or $-3$  C. $x = 3$ only  D. $x = -3$ only

43. If one root of $x^2 - x - 6 = 0$ is 3, the other root is:
     A. 2  B. $-2$  C. $-3$  D. 6

44. Solve: $2x^2 - 4x + 2 = 0$
     A. $x = 1$ (double root)  B. $x = \frac{1}{2}$ (double root)  C. $x = 2$ (double root)  D. $x = 0$

45. Solve: $x^2 + 7x + 12 = 0$ has already been covered (its product was question 41).
     For variety, assume Question 45 is used below in the surds section.

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SURDS (Questions 45–50)

45. Simplify: $\sqrt{50}$
     A. $5\sqrt{2}$  B. $2\sqrt{5}$  C. $25\sqrt{2}$  D. $10\sqrt{2}$

46. Simplify: $\sqrt{18} - \sqrt{8}$
     A. $\sqrt{2}$  B. $\sqrt{2} + \sqrt{3}$  C. $2\sqrt{2}$  D. $\sqrt{3}$

47. Rationalise the denominator of: $\frac{4}{\sqrt{3}}$
     A. $\frac{4\sqrt{3}}{3}$  B. $\frac{4\sqrt{3}}{2}$  C. $4\sqrt{3}$  D. $\frac{4}{3\sqrt{3}}$

48. Find $k$ if: $\sqrt{28} + \sqrt{112} - \sqrt{k} = \sqrt{175}$
     A. 7  B. 14  C. 49  D. 25

49. Expand and simplify: $(\sqrt{2} + \sqrt{3})^2$
     A. $5 + 2\sqrt{6}$  B. $5 + \sqrt{6}$  C. $2 + 3\sqrt{2}$  D. $2 + 3\sqrt{3}$

50. Simplify: $\frac{3\sqrt{2} + 2\sqrt{3}}{\sqrt{2}}$
     A. $3 + \sqrt{6}$  B. $3 + 2\sqrt{3}$  C. $3 + 2\sqrt{6}$  D. $2\sqrt{3} + 3$

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This 50‐question set is designed to test your knowledge and skills across these six topics in a style similar to JAMB exams. You can use these questions for practice and self-assessment.

Happy Studying!

Here are the correct answers to the 50 JAMB questions:

INDICES (Questions 1–10)

1. $2^3 \times 2^4 = 2^{3+4} = 2^7$ → B. $2^7$
2. $\frac{3^5}{3^2} = 3^{5-2} = 3^3$ → B. $3^3$
3. $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$ → A. $\frac{1}{25}$
4. $(2^3)^4 = 2^{3 \times 4} = 2^{12}$ → A. $2^{12}$
5. $a^2 = 16 \Rightarrow a = 4$, so $a^3 = 4^3 = 64$ → B. 64
6. $\frac{10^3}{10^5} = 10^{3-5} = 10^{-2}$ → B. $10^{-2}$
7. $4^2 \times 4^{-3} = 4^{2+(-3)} = 4^{-1}$ → A. $4^{-1}$
8. $2^x = 32 \Rightarrow 2^5 = 32$, so $x = 5$ → C. 5
9. $\frac{6^3}{6^3} = 1$ → A. 1
10. $(3^{-1})^{-2} = 3^{2}$ → A. $3^2$

LOGARITHMS (Questions 11–20)

11. $\log_2 8 = \log_2 (2^3) = 3$ → B. 3
12. $\log_{10} 100 = 2$ → B. 2
13. $\log_b (MN) = \log_b M + \log_b N$ → A. $\log_b M + \log_b N$
14. $\log_3 27 = \log_3 (3^3) = 3$ → B. 3
15. $\log_5 125 = \log_5 (5^3) = 3$ → B. 3
16. $\log_{10} x = 2 \Rightarrow x = 10^2 = 100$ → C. 100
17. $\log_a 1 = 0$ → A. 0
18. $\log_2 32 = 5$, so $\log_2 16 = 4$ → C. 4
19. $\log_b (1/M) = -\log_b M$ → A. $-\log_b M$
20. $\log_4 x = 3 \Rightarrow x = 4^3 = 64$ → A. 64

NUMBER BASES (Questions 21–30)

21. $1010_2 = 1(2^3) + 0(2^2) + 1(2^1) + 0(2^0) = 8 + 0 + 2 + 0 = 10$ → A. 10
22. $15_{10} = 1111_2$ → B. $1111_2$
23. $26_{10} = 1A_{16}$ → A. $1A$
24. $17_8 = 1(8^1) + 7(8^0) = 8 + 7 = 15$ → B. 15
25. $1101_2 = 1(2^3) + 1(2^2) + 0(2^1) + 1(2^0) = 8 + 4 + 0 + 1 = 13$ → C. 13
26. $20_{10} = 10100_2$ → A. $10100_2$
27. $31_{10} = 1F_{16}$ → A. $1F$
28. In base-8, 7 remains 7 → C. 7
29. The digit $A_{16} = 10_{10}$ → B. 10
30. $8_{10} = 20_4$ → A. $20_4$

PERCENTAGE ERROR (Questions 31–35)

31. $\frac{100 - 98}{100} \times 100 = 2\%$ → B. 2%
32. $\frac{50 - 45}{50} \times 100 = 10\%$ → C. 10%
33. $\frac{200 - 190}{200} \times 100 = 5\%$ → B. 5%
34. $\frac{3}{75} \times 100 = 4\%$ → B. 4%
35. $\frac{50 - 48}{50} \times 100 = 4\%$ → B. 4%

QUADRATIC EQUATIONS (Questions 36–45)

36. $x^2 - 5x + 6 = (x-2)(x-3)$, so $x = 2$ or $x = 3$ → A. 2 or 3
37. $2x^2 + 3x - 2 = (2x-1)(x+2)$, so $x = \frac{1}{2}$ or $x = -2$ → A. $\frac{1}{2}$ or -2
38. $x^2 + 4x + 4 = (x+2)^2$, so $x = -2$ (double root) → A. $-2$
39. Discriminant $= (-6)^2 - 4(1)(8) = 36 - 32 = 4$ → A. 4
40. $\sum$ of roots $= -\frac{-2}{3} = \frac{2}{3}$ → A. $\frac{2}{3}$
41. Product of roots $= 12$ → B. 12
42. $x^2 - 2x - 3 = (x-3)(x+1)$, so $x = -1$ or $x = 3$ → A. $-1$ or 3
43. If one root is 3, the other is $-\frac{-6}{3} = -2$ → B. -2
44. $2x^2 - 4x + 2 = (x-1)^2$, so $x = 1$ (double root) → A. $1$
45. $x^2 + 7x + 12 = (x+3)(x+4)$, so $x = -3$ or $x = -4$ → A. $-3$ or -4

SURDS (Questions 46–50)

46. $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ → A. $5\sqrt{2}$
47. $\sqrt{18} - \sqrt{8} = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2}$ → A. $\sqrt{2}$
48. $\frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$ → A. $\frac{4\sqrt{3}}{3}$
49. $\sqrt{28} + \sqrt{112} - \sqrt{k} = \sqrt{175}$, solving gives $k = 49$ → C. 49
50. $(\sqrt{2} + \sqrt{3})^2 = 2 + 3 + 2\sqrt{6} = 5 + 2\sqrt{6}$ → A. $5 + 2\sqrt{6}$

Let me know if you need explanations for any specific question.