Math topic中文

Expansion products

expansion of the product of algebraic expressions

Practice questions

  1. 1Factorise completely 9b2−6ab+a2−x29b^{2} - 6ab + a^{2} - x^{2}. Simplify 52x2−7x−4−84−x\dfrac{5}{2x^{2} - 7x - 4} - \dfrac{8}{4 - x}.
  2. 2先化简，再求值：(2x−y)(x−y)−(2x−y)2(2x-y)(x-y)-{{(2x-y)}^{2}}，其中x=−1,y=−2.x=-1,y=-2.
  3. 3(1)先化简，再求值：(a−b)2+b(a−b)(a-b)^{2}+b(a-b)，其中a=2a=2，b=−12b=- \dfrac {1}{2}． (2)先化简，再求值：(3x+2)(3x−2)−5x(x−1)−(2x−1)2(3x+2)(3x-2)-5x(x-1)-(2x-1)^{2}，其中x=−13x=- \dfrac {1}{3}…
  4. 4Simplify 3x2×4y53x^2 \times 4y^5. Answer ... 3(2x−1)−53(2x-1)-5. Answer ...
  5. 5Simplify −2(−3x−4)+5-2(-3x-4)+5. Answer ...
  6. 62x2−5xy−12y2x2−16y2\displaystyle \frac{2x^2-5xy-12y^2}{x^2-16y^2} Simplify. Answer ...
  7. 7A bag contains 12 balls. 2 of the balls are red, n are green balls and the rest are blue balls. Eunice takes two balls f…
  8. 8Simplify 3a25b3+(12c25ab)2\frac{3a^2}{5b^3} + \left(\frac{12c}{25ab}\right)^2. nn is a positive integer. Show that, for all nn, (3n+2)2−(3n−2)2(3n+2)^2 - (3n-2)^2…
  9. 9下列各式中正确的是()(\quad) A. a4⋅a2=a8a^{4}\cdot a^{2}=a^{8} B. (−2a2)3=−6a6(-2a^{2})^{3}=-6a^{6} C. 6a2+13a=2a+1\dfrac{6a^{2}+1}{3a}=2a+1 D. a(a−3)=a2−3aa(a-3)=a^{2}-3a…
  10. 10The sum, SnS_n, of the first nn terms of the sequence u1,u2,u3,…u_1, u_2, u_3, \dots is given by Sn=n2+4n.S_n = n^2 + 4n . Find unu_n…
  11. 11Factorise the following completely. 45b−18ab−2a2+5a45b - 18ab - 2a^2 + 5a 2p2−23p−162p^2 - \dfrac{2}{3}p - \dfrac{1}{6}
  12. 12Expand and simplify (2x+5)(3+4x)(2x+5)(3+4x).
  13. 13Write 92x+19^{2x+1} as a power of 3. Simplify 9a24b(27a3b6)−1\dfrac{9a^2}{4b}\left(\dfrac{27a^3b}{6}\right)^{-1}.
  14. 14Solve the equation 3−2x7=13(2x−1).\displaystyle \frac{3-2x}{7} = \frac{1}{3}(2x-1). Answer x = ... Simplify 25a2−9b215a−10ab+6b2−9b\displaystyle \frac{25a^{2}-9b^{2}}{15a-10ab+6b^{2}-9b}…
  15. 15Solve x/5+(2x−3)/4=−6x/5 + (2x-3)/4 = -6. Answer x = ... Simplify 2(p−3q)/(10r)+(3q−p)2/r2(p-3q)/(10r) + (3q-p)^2/r. Answer ... Solve the inequality x+1≤7x−24<5x+195x+1 \le \dfrac{7x-2}{4} < \dfrac{5x+19}{5}…
  16. 16The diagram shows a circle C1C_1 with centre P(5,3)P(5, 3). The circle touches the x-axis, the tangent OQ and another line Q…
  17. 17Town P and Town Q are 40 km apart. A car is travelling from Town P to Town Q at an average speed of x km/h. During its r…
  18. 18Factorise a3−2a2b−4a+8b.a^3 - 2a^2b - 4a + 8b. Solve 3x+29x=17x−3.\dfrac{3x+2}{9x} = \dfrac{1}{7x-3}.
  19. 19Solve the inequality −3x+42<5−x7\frac{-3x+4}{2} < \frac{5-x}{7}. Answer ... Simplify (64a6b−15)−13\left(\dfrac{64a^{6}}{b^{-15}}\right)^{-\tfrac{1}{3}}…
  20. 20Factorise x2+5x−6x^2+5x-6. Answer ... Hence, solve (p−1)2+5(p−1)−6=0(p-1)^2+5(p-1)-6=0. Answer p = ...
  21. 21Factorise completely. 24a2b+12ab2−ab24a^2b + 12ab^2 - ab Answer ... mn−18−9m+2nmn - 18 - 9m + 2n Answer ... Expand and simplify (−2x+3q)(x−2q)(-2x+3q)(x-2q)…
  22. 22先化简，再求值： [(x+y)(x−y)+2y(x−y)−(x−y)2]÷(2y)[(x+y)(x-y)+2y(x-y)-(x-y)^{2}]÷(2y)，其中x=1x=1，y=2y=2．
  23. 23Write down, and simplify the terms in the expansion of (2+x)5(2 + x)^5 in ascending powers of xx. Hence find the coefficien…
  24. 24Show that (3n+1)2+2(3n + 1)^2 + 2 is always divisible by 3 for all integer values of nn. Answer
  25. 25已知关于xx的一元二次方程(x−2)(x−5)−m2=0(x-2)(x-5)-m^{2}=0． (1)求证：对于任意实数mm，方程总有两个不相等的实数根； (2)若方程的一个根是11，求mm的值及方程的另一个根．
  26. 26先化简，再求值：a(a−4)+(1−a)(1+a)a(a-4)+(1-a)(1+a)，其中a=34a= \dfrac {3}{4}．
  27. 27Expand and simplify (3x−y)(y+x).(3x - y)(y + x). Factorise completely 3−12a23 - 12a^2.
  28. 28函数y=(x−2)(3−x)y=(x-2)(3-x)取得最大值时，x=x= ___ ．
  29. 29Alan drove a distance of xx km and his car used 7.5 litres of fuel. Write down an expression, in terms of xx, for the…
  30. 30Express 2x2−6x−122x^2 - 6x - 12 in the form k(x−a)2−bk(x - a)^2 - b, where kk is an integer. Answer: ... Hence, solve 2x2−6x−12=02x^2 - 6x - 12 = 0…
  31. 31Factorise a2b4−ab2.Answer...Expandandsimplifya^2b^4 - ab^2. Answer ... Expand and simplify (3x + 2)^2 + 4x(2 - x). Answer ...
  32. 32John and Peter took part in a marathon race. They each ran 42 km. (a) John ran at a constant speed of x kilometres per h…
  33. 33Old MacDonald has a farm with a rock wall around it. The figure shows a portion of his farm where he plans to build two…
  34. 34In the following sequence, (1×2)−2=0(1\times2)-2=0 (2×3)−4=2(2\times3)-4=2 (3×4)−6=6(3\times4)-6=6 (4×5)−8=12(4\times5)-8=12 ... (a×12)−b=c(a\times12)-b=c ..…
  35. 35The diagram shows a cylindrical container used to dispense coffee in a hotel. Calculate the volume of the cylinder. Answ…
  36. 36先化简，再求值：(5x−y)(y+2x)−(3y+2x)(3y−x)(5x-y)(y+2x)-(3y+2x)(3y-x)，其中x=1x=1，y=2y=2．
  37. 37(1)计算：(2x+1)(x−4)−(7−x)(−7−x)(2)(2x+1)(x-4)-(7-x)(-7-x)(2)先化简，再求值：(5m−4n2)2−(5m+4n2)2+10mn( \dfrac {5m-4n}{2})^{2}-( \dfrac {5m+4n}{2})^{2}+10mn，其中m=2m=2，n=−15n=- \dfrac {1}{5}…
  38. 38The figure shows the first three figures of a sequence formed by sticks of the same size. Find the number of small squar…
  39. 39计算或因式分解： (1)计算 [(a+b)  ⁣2+(a−b)  ⁣2](a  ⁣2−b  ⁣2)(3x  ⁣2−4x+5)(3x  ⁣2+4x−5)(2)[(a+b){\,\!}^{2}+(a-b){\,\!}^{2}](a{\,\!}^{2}-b{\,\!}^{2})(3x{\,\!}^{2}-4x+5)(3x{\,\!}^{2}+4x-5)(2)因式…
  40. 40The diagram below shows the sketch of the graph y = -x^2 + p x + q. The graph cuts the x-axis at points A and B(3, 0). T…
  41. 41计算(x2)3×(−2x)4(x^{2})^{3}×(-2x)^{4}的结果是()() A. 16x916x^{9} B. 16x1016x^{10} C. 16x1216x^{12} D. 16x2416x^{24}
  42. 42Given 3x−yx+2y=13\dfrac{3x - y}{x + 2y} = \dfrac{1}{3}, find the ratio x : y. Answer x : y = ... Solve the inequality 2−3x3<2x−16\dfrac{2 - 3x}{3} < \dfrac{2x - 1}{6}…
  43. 43Express 9−8x+x29 - 8x + x^2 in the form a+(x+b)2a + (x + b)^2. Find the value of aa and of bb. Answer a = ... b = ... Explain why…
  44. 44Solve the equation x3−3x−74=8\frac{x}{3} - \frac{3x-7}{4} = 8
  45. 45如果(x+2)(x−6)=x  ⁣2+px+q(x+2)(x-6)=x{\,\!}^{2}+px+q ，则p+q=p+q= ___
  46. 46下列各式从左到右的变形中，属于因式分解的是()(\quad) A. x(2a+1)=2ax+xx(2a+1)=2ax+x B. x2−2x+4=x(x−2)+4x^{2}-2x+4=x(x-2)+4 C. x2−36+9x=(x+6)(x−6)+9xx^{2}-36+9x=(x+6)(x-6)+9x D. m2−n2=(m−n)(m+n)m^{2}-n^{2}=(m-n)(m+n)…
  47. 47By using factorisation, solve (x - 3)(2x + 8) = -12. Answer: x = ... Solve the equation $$\frac{4}{2x-3} - \frac{3}{x+2} = 1.…
  48. 48Solve the inequalities 2x+13<4(x+2)≤x+412x + 13 < 4(x + 2) \le x + 41. Answer ___ Hence list all the prime integer values of x which sa…
  49. 49Factorise -2x^2 + x + 3. Answer ... Factorise completely 8x^3 - 18xy^2. Answer ...
  50. 50Expand and simplify a(5a−2b)(5a+2b)a(5a-2b)(5a+2b).
  51. 51The first four terms of a sequence are 5, 94\dfrac{9}{4}, 139\dfrac{13}{9}, 1716\dfrac{17}{16}. State the fifth term of the…
  52. 52Using a separate diagram for each part, represent on the number line the solution set of 3(5−x)>x+133(5 - x) > x + 13 x2−12x<14(3−2x)x^{2} - \frac{1}{2}x < \frac{1}{4}(3 - 2x)…
  53. 53A supermarket sold all of its toilet rolls at a price of $xx per pack in January. The revenue made from selling the packs of toilet rolls in January was $5940. Write down an expression in x, for the number of packets of toilet rolls sold in January. Answer ... In February, the supermarket ordered an additional 600 packs to the number sold in January and sold them at 50 cents more per pack. Write down an expression in x, for the total amount of money received in dollars, if all the packs of toilet rolls were sold in February. Answer $ ... The supermarket received $3870 more from the sales of toilet rolls in February as compared to January. Write down an equation in x to represent this information, and show that it reduces to $$…
  54. 54下列各式从左到右的变形中，属于因式分解的是(    )(\:\:\:\:) A. 6x+9y+3=3(2x+3y)6x+9y+3=3(2x+3y) B. x2−1=(x−1)2x ^{2} -1=(x-1) ^{2} C. (x+y)2=x2+2xy+y2(x+y) ^{2} =x ^{2} +2xy+y ^{2}…
  55. 55已知αα、ββ是一元二次方程x2−2021x+2020=0x^{2}-2021x+2020=0的两实根，则代数式(α−2021)(β−2021)=(α-2021)(β-2021)= ___ .
  56. 56The variables x and y are connected by the equation y=13x(11−x2).Somecorrespondingvaluesofxandy,correctto1decimalplaces,aregiveninthefollowingtable.Calculatethevalueofp.Answerp=...Onthegridopposite,drawthegraphofy = \frac{1}{3}x(11 - x^2). Some corresponding values of x and y, correct to 1 decimal places, are given in the following table. Calculate the value of p. Answer p = ... On the grid opposite, draw the graph of …
  57. 57The graphs of y=a(x+2)(4−x)y = a(x+2)(4-x) and y=−12x+52y = -\frac{1}{2}x + \frac{5}{2} are drawn on the grid. Write down the equation of…
  58. 58计算： (1)(a2)3⋅(a2−2ab+1);(2)6x(−x2−xy+y2)(−xy);(3)2x2y(3−x4y)−(5x3y)2;(4)x2(x−1)−x(x2+x−1);(5)(−2xy2)2⋅(14y2−12x2−32xy);(6)3a(2a2−4a+3)−2a2(3a−4).(1)(a^{2})^{3}\cdot(a^{2}-2ab+1);(2)6x(-x^{2}-xy+y^{2})(-xy);(3)2x^{2}y(3-x^{4}y)-(5x^{3}y)^{2};(4)x^{2}(x-1)-x(x^{2}+x-1);(5)(-2xy^{2})^{2}\cdot(\dfrac{1}{4}y^{2}-\dfrac{1}{2}x^{2}-\dfrac{3}{2}xy);(6)3a(2a^{2}-4a+3)-2a^{2}(3a-4).…
  59. 59单项式与多项式相乘．结果仍是 ，结果的项数等于 的项数．
  60. 60a=3−bb+4+2a = \frac{3 - b}{b + 4} + 2 Find a when b = -6. Answer a = ... Rearrange the formula to make b the subject. Answer b =…