Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 394 · 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 + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
2. 2.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 4899 (d) 5041
3. 3.If a + b = 8 and ab = 12, what is (a − b)²? (a) 4 (b) 64 (c) 40 (d) 16
4. 4.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 32 (d) 33
5. 5.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
9. 9.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 11 (c) 10 (d) 30
10. 10.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
11. 11.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 229 (d) 139
12. 12.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 961 (d) 851
13. 13.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 34 (d) 4
14. 14.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
15. 15.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
16. 16.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 441 (d) 9
19. 19.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 9 (d) 8
20. 20.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
22. 22.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 289 (c) 217 (d) 1
23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3844 (d) 3596
25. 25.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
26. 26.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 289 (d) 25
27. 27.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4634 (d) 4896
28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
29. 29.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 1 and 24 (d) 3 and 8
30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9604 (c) 10404 (d) 9996
31. 31.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 9 (c) 7 (d) 5
32. 32.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 4 (d) 100
33. 33.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 2035 (d) 1225
34. 34.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
36. 36.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1764 (d) 1596
37. 37.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73
38. 38.If a + b = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 7 (d) 49
39. 39.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025

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