Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 838 · 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.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 441 (d) 361
2. 2.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
4. 4.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 20 (d) 36
5. 5.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)
6. 6.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 10 (d) 12
7. 7.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
9. 9.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 3 (d) 9
10. 10.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 219 (c) 289 (d) 149
11. 11.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 147 (c) 169 (d) 125
12. 12.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9226 (d) 9984
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
14. 14.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
17. 17.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
18. 18.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5476 (c) 4356 (d) 5486
19. 19.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
20. 20.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 5041 (d) 4899
21. 21.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 484 (d) 404
22. 22.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
23. 23.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 6 (c) 146 (d) 36
24. 24.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 37 (c) 35 (d) 13
25. 25.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
26. 26.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
27. 27.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
28. 28.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 8 (c) 11 (d) 12
29. 29.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 625 (c) 425 (d) 635
30. 30.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 37 (c) 49 (d) 43
31. 31.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 6 (c) 36 (d) 50
32. 32.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
33. 33.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
34. 34.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
35. 35.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 7921 (d) 8291
36. 36.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
38. 38.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
39. 39.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
40. 40.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609

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