Algebraic identities
Class 8 Maths · Algebraic expressions and identities · 36 questions with answers
There are only three identities in Class 8 and memorising them takes ten minutes. What the paper tests is recognition - seeing that x² - 49 is the third one in disguise, or that 103 × 97 can be done in one line. That recognition only comes from doing enough of them, which is why this is the chapter where practice genuinely cannot be substituted.
- Expands using each of the three standard identities
- Uses an identity to do arithmetic mentally
- Recognises an identity inside a question that does not announce it
- Rearranges an identity to find an unknown expression
Exercise 1 of 3 · 12 questions
Expand using an identity. Name the one you used.
Write the answer · Warm-up
Here’s one done for you
(2x + 3)² = 4x² + 12x + 9. The trap is writing 2x² instead of 4x² - the whole of 2x is being squared, not just the x. Whenever the first term has a number in front of it, square the number too. That single slip accounts for more lost marks in this chapter than any misremembered identity.
- 1)(x + 5)²
- 2)(a - 7)²
- 3)(2x + 3)²
- 4)(3m - 4)²
- 5)(x + 6)(x - 6)
- 6)(2y + 5)(2y - 5)
- 7)(x + 9)²
- 8)(a - 5)²
- 9)(3x + 7)²
- 10)(5m - 2)²
- 11)(x + 8)(x - 8)
- 12)(3y + 4)(3y - 4)
Answers
- 1) x² + 10x + 25, using (a + b)² = a² + 2ab + b².
- 2) a² - 14a + 49, using (a - b)² = a² - 2ab + b².
- 3) 4x² + 12x + 9. Here a = 2x, so a² = 4x² and 2ab = 2 × 2x × 3 = 12x.
- 4) 9m² - 24m + 16. Here a = 3m, so a² = 9m² and 2ab = 2 × 3m × 4 = 24m.
- 5) x² - 36, using (a + b)(a - b) = a² - b².
- 6) 4y² - 25. Here a = 2y, so a² = 4y².
- 7) x² + 18x + 81. Identity (a + b)² = a² + 2ab + b².
- 8) a² - 10a + 25. Identity (a - b)² = a² - 2ab + b².
- 9) 9x² + 42x + 49. Identity (a + b)², with a = 3x and b = 7.
- 10) 25m² - 20m + 4. Identity (a - b)², with a = 5m and b = 2.
- 11) x² - 64. Identity (a + b)(a - b) = a² - b².
- 12) 9y² - 16. Identity (a + b)(a - b), with a = 3y and b = 4.
Exercise 2 of 3 · 12 questions
Work these out using an identity. No long multiplication.
Write the answer · Guided
Here’s one done for you
103 × 97 = 9991, because 103 = 100 + 3 and 97 = 100 - 3, so the answer is 100² - 3² = 10000 - 9. This exercise exists for one reason: it turns three abstract formulas into something obviously useful. A child who can do 103 × 97 in their head stops treating identities as things to be memorised for a test.
- 1)102²
- 2)98²
- 3)103 × 97
- 4)51²
- 5)995 × 1005
- 6)49 × 51
- 7)101²
- 8)97²
- 9)104 × 96
- 10)52²
- 11)998 × 1002
- 12)48 × 52
Answers
- 1) 10404. (100 + 2)² = 10000 + 400 + 4.
- 2) 9604. (100 - 2)² = 10000 - 400 + 4.
- 3) 9991. (100 + 3)(100 - 3) = 10000 - 9.
- 4) 2601. (50 + 1)² = 2500 + 100 + 1.
- 5) 999975. (1000 - 5)(1000 + 5) = 1000000 - 25.
- 6) 2499. (50 - 1)(50 + 1) = 2500 - 1.
- 7) 10,201. (100 + 1)² = 10000 + 200 + 1.
- 8) 9,409. (100 - 3)² = 10000 - 600 + 9.
- 9) 9,984. (100 + 4)(100 - 4) = 10000 - 16.
- 10) 2,704. (50 + 2)² = 2500 + 200 + 4.
- 11) 999,996. (1000 - 2)(1000 + 2) = 1000000 - 4.
- 12) 2,496. (50 - 2)(50 + 2) = 2500 - 4.
Exercise 3 of 3 · 12 questions
Expand or evaluate. Some need an identity, some need rearranging.
Write the answer · Practice
Here’s one done for you
If a + b = 7 and ab = 12, then a² + b² = 25. You cannot find a and b separately from this - and you do not need to. Square the sum: (a + b)² = a² + 2ab + b², so 49 = a² + b² + 24, giving 25. This is the question type that separates memorising the identity from understanding it, and it appears in some form in every paper from here to Class 10.
- 1)(x + 3)(x + 5)
- 2)(x - 4)(x + 7)
- 3)(2a + 3)(a - 5)
- 4)If a + b = 7 and ab = 12, find a² + b².
- 5)If x + 1/x = 5, find x² + 1/x².
- 6)Simplify (x + y)² - (x - y)².
- 7)(x + 2)(x + 9)
- 8)(x - 3)(x + 8)
- 9)(3a + 2)(a - 4)
- 10)If a + b = 9 and ab = 20, find a² + b².
- 11)If x + 1/x = 6, find x² + 1/x².
- 12)Simplify (x + y)² + (x - y)².
Answers
- 1) x² + 8x + 15.
- 2) x² + 3x - 28.
- 3) 2a² - 7a - 15. Expanding gives 2a² - 10a + 3a - 15.
- 4) 25. Since (a + b)² = a² + 2ab + b², we get a² + b² = 49 - 2(12) = 25.
- 5) 23. Squaring both sides gives x² + 2 + 1/x² = 25, so x² + 1/x² = 23.
- 6) 4xy. The first expands to x² + 2xy + y² and the second to x² - 2xy + y²; subtracting leaves 4xy.
- 7) x² + 11x + 18.
- 8) x² + 5x - 24.
- 9) 3a² - 10a - 8.
- 10) a² + b² = (a + b)² - 2ab = 81 - 40 = 41.
- 11) x² + 1/x² = (x + 1/x)² - 2 = 36 - 2 = 34.
- 12) 2x² + 2y². The cross terms cancel because one is +2xy and the other -2xy.
While your child works
- When the first term has a number in front - (2x + 3)² - check they squared the number too. That single slip is the commonest error in the chapter.
- The mental arithmetic exercise is the one that makes identities feel worth knowing. Do not skip it if time is short.
- The last two questions cannot be done by memorising. If those come out, the chapter is genuinely understood.