Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 370 · 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 + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
2. 2.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 324 (d) 170
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6251 (d) 6399
6. 6.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)
7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 156 (d) 196
8. 8.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
9. 9.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
11. 11.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 11 (d) 13
12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
13. 13.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 37 (c) 13 (d) 36
14. 14.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
15. 15.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
16. 16.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
18. 18.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
19. 19.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 2 and 10 (c) 1 and 35 (d) 5 and 7
20. 20.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 121 (d) 181
21. 21.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6724 (d) 6396
22. 22.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
23. 23.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 7 (d) 49
24. 24.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
25. 25.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 50 (d) 6
26. 26.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 9 (d) 3
27. 27.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 41 (c) 13 (d) 43
29. 29.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 169 (c) 1 (d) 85
30. 30.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
32. 32.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
33. 33.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 5786 (c) 7056 (d) 6384
34. 34.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1946 (c) 1296 (d) 1616
35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
36. 36.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
37. 37.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 144 (d) 74
38. 38.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 7 and 8 (d) 1 and 56
39. 39.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8659 (c) 8109 (d) 7569
40. 40.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 85 (c) 121 (d) 103

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