Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 57 · 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.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
3. 3.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 32 (d) 1 and 11
4. 4.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
5. 5.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 9 (c) 7 (d) 8
6. 6.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 2 and 24 (d) 6 and 8
7. 7.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
8. 8.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
10. 10.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)
11. 11.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9614 (c) 9996 (d) 9604
12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 121 (d) 141
13. 13.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 31 (c) 29 (d) 11
14. 14.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6404 (d) 6734
15. 15.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
16. 16.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
17. 17.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1681 (d) 1599
20. 20.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
21. 21.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10816 (c) 10826 (d) 9216
22. 22.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 4 and 6 (d) 1 and 9
23. 23.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225
24. 24.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 11025 (c) 9025 (d) 9035
25. 25.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
26. 26.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)
27. 27.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3374 (d) 3844
28. 28.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 12 (c) 13 (d) 40
29. 29.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 3 and 4 (c) 1 and 10 (d) 2 and 5
30. 30.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)
31. 31.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 625 (d) 225
32. 32.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
33. 33.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 253 (d) 85
34. 34.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 4 (c) 10 (d) 2
35. 35.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)
36. 36.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 8 (c) 2 and 4 (d) 1 and 5
37. 37.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
39. 39.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
40. 40.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36

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