15. Algebraic identities and simplification
Free study material · concepts, shortcuts & solved questions
Memorise identities as tools: (a+b)²=a²+2ab+b²; (a−b)²=a²−2ab+b²; a²−b²=(a−b)(a+b); (a+b)³=a³+3a²b+3ab²+b³. Factor before expanding when that is shorter.
Method before formula Signs matter. A common mistake is writing +2ab in (a−b)². Substitute a convenient base such as 100 when calculating squares near a round number. |
Formula and decision table
Example | Method |
103² | (100+3)²=10000+600+9=10609 |
99² | (100−1)²=9801 |
x²−25 | (x−5)(x+5) |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
103² is: | Use (100+3)². | B. 10609 |
a²−b² factors as: | Difference of squares is a product. | C. (a−b)(a+b) |
Middle term in (a−b)² is: | The sign is negative. | B. −2ab |
What an examiner is testing
The options around Algebraic identities and simplification usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: 103² is: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 10609, because Use (100+3)².
Example 2: a²−b² factors as: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is (a−b)(a+b), because Difference of squares is a product.
Example 3: Middle term in (a−b)² is: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is −2ab, because The sign is negative.
Chapter practice
1. 103² is:
(A) 10509 (B) 10609 (C) 10709 (D) 10809
Answer: B. 10609 | Explanation: Use (100+3)².
2. a²−b² factors as:
(A) (a−b)² (B) (a+b)² (C) (a−b)(a+b) (D) a²+b²
Answer: C. (a−b)(a+b) | Explanation: Difference of squares is a product.
3. Middle term in (a−b)² is:
(A) +2ab (B) −2ab (C) 0 (D) ab²
Answer: B. −2ab | Explanation: The sign is negative.
SSC CGL speed and trap clinic
For Algebraic identities and simplification, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Algebraic identities and simplification? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Algebraic identities and simplification question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |