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Algebraic Identities & Elementary Surds

Quantitative Aptitude · Algebra, Equations & Graphs

Useful for: SSC CGL

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Overview

Combining identities with surd rules simplifies expressions like (√a + √b)² or 1/(√a − √b) in one step.

Key points

(√a + √b)(√a − √b) = a − b.
(√a + √b)² = a + b + 2√(ab).
Rationalising: multiply numerator and denominator by the conjugate.
If x = a + √b, then 1/x can be found by rationalising.
Comparing surds: square both sides when both are positive.

Explanation

x = √3 + 1 → x² = 4 + 2√3.
For x + 1/x with x = 2 + √3: 1/x = 2 − √3, so x + 1/x = 4.
√(a + 2√b) can be written as √m + √n when m + n = a and mn = b.
To compare √3 + √5 and √2 + √6, square both: 8 + 2√15 vs 8 + 2√12.

Step by step

Spot the conjugate or identity.
Rationalise or expand.
Simplify and approximate to check.

Examples

(√5 + √3)(√5 − √3) = 5 − 3 = 2.
6/(√5 − √2) = 6(√5 + √2)/(5 − 2) = 2(√5 + √2).
x = 2 + √3 → x + 1/x = (2 + √3) + (2 − √3) = 4 → x² + 1/x² = 16 − 2 = 14.

Traps and advanced points

Trap: Writing √(a + b) as √a + √b.
Trap: Rationalising with the same sign instead of the conjugate.
Trap: Dropping the 2√(ab) term when squaring.

Quick revision

Conjugate removes the root.
x = 2 + √3 → x + 1/x = 4.
Square both sides to compare surds.

Practice MCQs

If a + b = 5 and ab = 6, then a² + b² is
1. 13
2. 25
3. 19
4. 11
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Answer: (a) 13. (a + b)² − 2ab = 25 − 12.
If x + 1/x = 3, then x² + 1/x² is
1. 7
2. 9
3. 11
4. 5
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Answer: (a) 7. (x + 1/x)² − 2 = 9 − 2.
The value of (√3 + √2)(√3 − √2) is
1. 1
2. 5
3. √6
4. 0
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Answer: (a) 1. (a + b)(a − b) = a² − b² = 3 − 2.
Rationalising 1/(√5 − 2) gives
1. √5 − 2
2. √5 + 2
3. (√5 + 2)/9
4. 1/3
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Answer: (b) √5 + 2. Multiply by √5 + 2; the denominator becomes 5 − 4 = 1.
If a + b = 4 and ab = 3, then a³ + b³ is
1. 28
2. 64
3. 36
4. 52
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Answer: (a) 28. (a + b)³ − 3ab(a + b) = 64 − 36.
√48 in simplest form is
1. 6√2
2. 3√4
3. 2√12
4. 4√3
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Answer: (d) 4√3. 48 = 16 × 3.

More topics in this chapter

Linear Equations & Graphs

Notes and questions are written by Kundali IQ for practice. They are not official papers; always check the latest official notification for the syllabus and pattern.