Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 25 · 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 + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
2. 2.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
3. 3.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 9984 (d) 10816
4. 4.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2491 (d) 2219
5. 5.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 3616 (c) 4106 (d) 4096
6. 6.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 53 (c) 15 (d) 54
7. 7.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2809 (d) 2509
8. 8.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
10. 10.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 19 (d) 9
11. 11.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 5776 (c) 7056 (d) 7066
12. 12.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
13. 13.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 14 (d) 13
14. 14.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 146 (c) 366 (d) 256
15. 15.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
16. 16.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9604 (d) 9996
17. 17.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 429 (d) 219
18. 18.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 29 (d) 31
19. 19.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 17 (c) 11 (d) 18
20. 20.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
21. 21.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 144 (d) 2
22. 22.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)
23. 23.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8101 (c) 8281 (d) 7921
24. 24.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
26. 26.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)
27. 27.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 34 (c) 94 (d) 49
28. 28.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
30. 30.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
31. 31.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6399 (d) 6251
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
33. 33.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 37 (d) 36
34. 34.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
35. 35.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 235 (c) 225 (d) 375
36. 36.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 132 (d) 68
37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
38. 38.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 11 (d) 9
39. 39.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 160 (d) 352
40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181

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