Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 399 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 333 (d) 117
3. 3.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851
4. 4.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
5. 5.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
6. 6.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 1575 (c) 1235 (d) 2025
7. 7.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 262 (d) 196
8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
10. 10.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10826 (b) 9216 (c) 10816 (d) 10016
11. 11.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 104 (d) 144
12. 12.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 31 (c) 36 (d) 26
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
14. 14.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
15. 15.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 5041 (c) 4771 (d) 4899
16. 16.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 25 (c) 17 (d) 33
17. 17.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
19. 19.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 12 (c) 10 (d) 30
20. 20.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 64 (c) 16 (d) 62
21. 21.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 400 (d) 16
22. 22.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56
23. 23.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 10404 (d) 9604
24. 24.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 3 and 8 (d) 1 and 10
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
27. 27.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 739 (c) 1089 (d) 891
28. 28.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 74 (d) 109
29. 29.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 122 (d) 166
30. 30.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
33. 33.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 676 (d) 1156
34. 34.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3364 (c) 3596 (d) 3374
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
37. 37.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 117 (d) 45
38. 38.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
39. 39.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 12 (d) 32
40. 40.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 16 (d) 324

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