Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 58 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
2. 2.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 8 (d) 15
4. 4.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 196 (c) 10 (d) 100
5. 5.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 12 (d) 1 and 42
6. 6.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
7. 7.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
8. 8.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
9. 9.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
10. 10.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
12. 12.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
13. 13.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 36 (d) 13
14. 14.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
15. 15.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 10 (d) 12
16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
18. 18.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
19. 19.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5625 (c) 7225 (d) 5635
20. 20.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 11 (d) 18
21. 21.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 81 (d) 9
22. 22.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
23. 23.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 6 (d) 10
24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 169 (c) 1 (d) 225
25. 25.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 15 (c) 3 and 21 (d) 1 and 63
26. 26.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 7569 (c) 7579 (d) 8091
27. 27.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
29. 29.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 2 (d) 10
30. 30.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 4 (c) 16 (d) 40
31. 31.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
32. 32.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2209 (d) 2219
33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
34. 34.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9996 (c) 9604 (d) 9614
35. 35.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
37. 37.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 2 (c) 4 (d) 64
38. 38.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
39. 39.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 11 (d) 16
40. 40.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13

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