Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 93 · 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 = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
3. 3.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
4. 4.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3979 (c) 3609 (d) 3969
5. 5.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
6. 6.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 7 (c) 169 (d) 289
7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 116 (d) 196
8. 8.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2809 (c) 2209 (d) 2819
10. 10.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
11. 11.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 36 (c) 324 (d) 6
12. 12.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 6 and 8 (d) 1 and 13
13. 13.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 15 (d) 14
14. 14.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
16. 16.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
17. 17.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2704 (d) 2714
18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
19. 19.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
20. 20.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
22. 22.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 61 (d) 181
23. 23.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
24. 24.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 121 (c) 141 (d) 111
25. 25.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 256 (d) 576
28. 28.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
29. 29.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
30. 30.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
31. 31.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
32. 32.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
33. 33.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
34. 34.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 7 (d) 4
35. 35.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
36. 36.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
37. 37.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 43 (d) 13
38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
40. 40.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 53 (c) 81 (d) 67

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