Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 890 · 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.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 25 (d) 24
2. 2.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
4. 4.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
5. 5.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 10 (c) 148 (d) 196
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
7. 7.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 6889 (d) 5929
8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
9. 9.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 8 (c) 3 and 6 (d) 1 and 14
10. 10.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9614 (c) 9604 (d) 10404
11. 11.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 16 (c) 4 (d) 40
12. 12.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 484 (d) 164
13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
14. 14.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
17. 17.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 122 (c) 166 (d) 133
18. 18.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 7 and 9 (c) 3 and 21 (d) 1 and 15
19. 19.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
20. 20.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 429 (d) 289
21. 21.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
22. 22.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
23. 23.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 18 (d) 19
24. 24.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
25. 25.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 9 (d) 7
26. 26.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 2 and 10 (d) 1 and 11
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
28. 28.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 361 (c) 277 (d) 529
29. 29.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
30. 30.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
31. 31.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)
32. 32.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 7 (c) 11 (d) 10
33. 33.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 9 (d) 10
34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
36. 36.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
37. 37.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
38. 38.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 169 (c) 97 (d) 133
39. 39.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 55 (d) 15
40. 40.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 25 (d) 361

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