Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 928 · 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.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 1 and 12 (d) 2 and 18
2. 2.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
3. 3.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
5. 5.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 11 (c) 13 (d) 35
6. 6.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 9 (d) 81
7. 7.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
8. 8.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 16 (d) 3 and 7
9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 217 (c) 433 (d) 289
11. 11.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 144 (d) 104
12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
13. 13.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 141 (d) 111
14. 14.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
15. 15.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
16. 16.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 7 (d) 6
17. 17.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)
18. 18.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 1089 (d) 739
19. 19.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 1 (d) 41
21. 21.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 21 (d) 9
22. 22.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 16 (d) 15
23. 23.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
24. 24.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
25. 25.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
26. 26.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 2475 (c) 2035 (d) 3025
27. 27.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3721 (d) 3601
28. 28.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 76 (c) 100 (d) 148
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
30. 30.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
31. 31.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
32. 32.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 324 (c) 164 (d) 2
33. 33.If a + b = 14 and ab = 33, what is a² + b²? (a) 130 (b) 163 (c) 262 (d) 196
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
35. 35.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 8075 (d) 7225
36. 36.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 16 (d) 10
37. 37.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 5 (c) 25 (d) 49
38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
40. 40.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776

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