Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 338 · 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.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
2. 2.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
3. 3.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 4761 (d) 5041
4. 4.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2496 (d) 2304
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
6. 6.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
8. 8.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 1225 (d) 2025
9. 9.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
10. 10.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
11. 11.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 5184 (d) 4634
12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 101 (d) 141
13. 13.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
14. 14.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
15. 15.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 529 (c) 299 (d) 391
16. 16.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 148 (d) 52
17. 17.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 37 (d) 35
18. 18.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)
19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
20. 20.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
22. 22.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 625 (d) 875
23. 23.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 5041 (d) 4899
24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
25. 25.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 16 (d) 17
26. 26.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
27. 27.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 181 (d) 121
28. 28.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
29. 29.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 10016 (c) 10816 (d) 9216
30. 30.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3575 (c) 3025 (d) 3035
31. 31.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 73 (c) 25 (d) 121
32. 32.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 1 and 15 (d) 7 and 9
33. 33.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
34. 34.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
36. 36.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
38. 38.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
40. 40.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 16 (d) 58

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