Math topic中文

Add subtract fractions

addition and subtraction of algebraic fractions with linear or quadratic denominator such as: 1x−2+2x−3\dfrac{1}{x - 2} + \dfrac{2}{x - 3}; 1x2−9+2x−3\quad \dfrac{1}{x^{2} - 9} + \dfrac{2}{x - 3}; 1x−3+2(x−3)2\quad \dfrac{1}{x - 3} + \dfrac{2}{(x - 3)^{2}}

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：(xx−2−xx+2)÷4xx−2( \dfrac {x}{x-2}- \dfrac {x}{x+2})÷ \dfrac {4x}{x-2}，其中x=2016x=2016．
  3. 3张老板经营两种茶叶：第一种为“采花芽尖”，卖出aa千克总售价为bb元；第二种茶叶为“云顶雾绿”，比第一种茶叶多卖cc千克时，总售价比第一种多cc元。请你帮张老板计算一下，哪一种茶叶单价稍贵些？ ((其中b>a)b>a)…
  4. 4下列计算正确的是 ()(\quad) A. 1x+12x=13x\dfrac{1}{x}+\dfrac{1}{2x}=\dfrac{1}{3x} B. 1x−1y=1x−y\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{x-y} C.…
  5. 5Adam has some 50-cent and 20-cent coins in his wallet. Given that he has a total of 73 coins adding up to a value of $2…
  6. 6已知数列{an}\{a_{n}\}中，a1=76a_{1}=\dfrac{7}{6}，an2−an+1=an+1(n∈N∗)a_{n}^{2}-a_{n}+1=a_{n+1}(n\in N^{*})，SnS_{n}是{1an}\{\dfrac{1}{a_{n}}\}的前…
  7. 7Simplify 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…
  8. 8先化简，再求值：x2−2x+1x2+3x÷(1−4x+3)\frac { x ^ { 2 } -2x+1 } { x ^ { 2 } +3x }÷\left ( { 1-\frac { 4 } { x+3 } } \right )，然后从−3-3，0，1，3中选…
  9. 9有一列按一定顺序和规律排列的数： 第11个数是11×2\dfrac{1}{1×2}；第22个数是12×3\dfrac{1}{2×3}；第33个数是13×4… \dfrac{1}{3×4} …对任何正整数nn，第…
  10. 10a+2−42−a=a+2- \dfrac {4}{2-a}= ___ ．
  11. 11已知函数f(x)=2xx+1f(x)=\frac { 2x } { x+1 }． （1）用定义证明函数f(x)f(x)在区间(1，+∞)(1，+∞)上单调递增； （2）对任意x∈[2，4]x∈[2，4]都有f(x)≤mf(x)≤m成立，求实数mm…
  12. 12化简(a+3a−4a−3)(1a−2−1)\left(a+\dfrac{3a-4}{a-3}\right)\left(\dfrac{1}{a-2}-1\right)的结果等于 ().(\quad). A. −a−2-a-2 B. a−2a−3\dfrac{a-2}{a-3}…
  13. 13若1a−1b=4 \dfrac {1}{a}- \dfrac {1}{b}=4，则a−2ab−b2a+7ab−2b \dfrac {a-2ab-b}{2a+7ab-2b}的值是 ___ ．
  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. 15Write as a single fraction in its simplest form
  16. 16Solve 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}…
  17. 17Solve the inequality 4x+13>3−2x5\frac{4x+1}{3} > \frac{3-2x}{5}. Simplify (16a12b8)−14\left(\dfrac{16a^{12}}{b^8}\right)^{-\tfrac{1}{4}}, l…
  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. 20The variables x and y are connected by the equation y=x+5x2−3y = x + \frac{5}{x^2} - 3 Some corresponding values of x and y are…
  21. 21计算12a2−9+23−a\dfrac{12}{{{a}^{2}}-9}+\dfrac{2}{3-a}的结果是___.
  22. 22已知x1x_{1}、x2x_{2}是方程2x2−3x−1=02x^{2}-3x-1=0的两根，则1x1+1x2= ()\dfrac{1}{x_{1}}+\dfrac{1}{x_{2}}=\:(\quad) A. −1-1 B. 11 C. −3-3…
  23. 23Here are the first four terms of a sequence. 1351010715420\displaystyle \frac{13}{5}\quad \frac{10}{10}\quad \frac{7}{15}\quad \frac{4}{20}…
  24. 24Express x2−6x−2x^2-6x-2 in the form (x−a)2+b(x-a)^2 + b where aa and bb are constants. Answer ___ Hence, state the minimum value…
  25. 25Express 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…
  26. 26(1)解方程：13x−4−10x−3=4x−5−1x−1(2) \dfrac{13}{x-4}- \dfrac{10}{x-3}= \dfrac{4}{x-5}- \dfrac{1}{x-1} (2)已知a2+4a+1=0a^{2}+4a+1=0，且，a4+ma2+13a2+ma2+3a=5\dfrac{{a}^{4}+m{a}^{2}+1}{3{a}^{2}+m{a}^{2}+3a}=5…
  27. 27计算：2xx2−y2−2yx2−y2=\dfrac{2x}{{{x}^{2}}-{{y}^{2}}}-\dfrac{2y}{{{x}^{2}}-{{y}^{2}}}= ．
  28. 28Given 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}…
  29. 29Write as a single fraction in its simplest form. 3t2w⋅9t2w3\dfrac{3t^2}{w} \cdot \dfrac{9t^2}{w^3} Answer ----------------------…
  30. 30计算： (1)1x−3−13+x(2)m−1+2m−6m2−9÷2m+2m+3(1) \dfrac {1}{x-3}- \dfrac {1}{3+x}(2)m-1+ \dfrac {2m-6}{m^{2}-9}÷ \dfrac {2m+2}{m+3}．
  31. 31Factorise completely m2−2mn+n2−p2m^2 - 2mn + n^2 - p^2. Answer ... Express as a single fraction in its simplest form 72x−3+x+16−4x\dfrac{7}{2x-3} + \dfrac{x+1}{6-4x}…
  32. 32化简2x2−4−12x−4 \dfrac {2}{x^{2}-4}- \dfrac {1}{2x-4}．
  33. 33先化简，再求值： ​xx2−2x+1÷(x+1x2−1+1)\dfrac{x}{{x}^{2}-2x+1}÷( \dfrac{x+1}{{x}^{2}-1}+1) ，其中 x=2+1.x= \sqrt{2}+1 .
  34. 34设数列{an}\{a_{n}\}满足an+1=4an−2an+1(n∈N∗)(1)a_{n+1}= \dfrac {4a_{n}-2}{a_{n}+1}(n∈N^{*})(1)若a1=3a_{1}=3，bn=2−anan−1(n∈N∗)b_{n}= \dfrac {2-a_{n}}{a_{n}-1}(n∈N^{*})…
  35. 35Simplify 25m2−44−4m−15m2\dfrac{25m^2-4}{4-4m-15m^2}. Given that r=3−p24−qr = 3 - \sqrt{\dfrac{p^2}{4} - q}, evaluate rr when p=−8p = -8 and q=7q = 7…
  36. 36符号“∣abcd∣\left|\begin{array}{cc}a&b\\ c&d\end{array}\right|”称为二阶行列式，规定它的运算法则为∣abcd∣=ad−bc.\left|\begin{array}{cc}a&b\\ c&d\end{array}\right|=ad-bc.…
  37. 37Write as a single fraction in its simplest form 3(1−2h)2+82h−1\dfrac{3}{(1-2h)^2}+\dfrac{8}{2h-1}. Answer ...
  38. 38计算： 1a−2+2a+1−2a−1−1a+2\ \dfrac{1}{a-2}+\dfrac{2}{a+1}-\dfrac{2}{a-1}-\dfrac{1}{a+2}的值为()(\quad) A.  20(a2−4)(a2−1)\ \dfrac{20}{(a^{2}-4){(a}^{2}-1)}…
  39. 39a=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 =…
  40. 40Write as a single fraction in its simplest form −4x2−4+1x−2\displaystyle \frac{-4}{x^2-4} + \frac{1}{x-2}. Answer ...
  41. 41Write as a single fraction in the simplest form 3x2x2−50−1x−5\frac{3x}{2x^2-50}-\frac{1}{x-5}. x=25y7z+23x = \sqrt[3]{\frac{25y}{7z+2}} Fin…
  42. 42Write as a single fraction in its simplest form 31−x2−2x+1\dfrac{3}{1-x^2}-\dfrac{2}{x+1}. Answer ...
  43. 43Simplify 16a6b25−8a325ab3\displaystyle\frac{16a^6 b^2}{5} - \displaystyle\frac{8a^3}{25ab^3}. Answer ___ (i) Express −4x+x2−6-4x + x^2 - 6 i…
  44. 44Rearrange the formula 1a−12b=13c\dfrac{1}{a}-\dfrac{1}{2b}=\dfrac{1}{3c} to make bb the subject. Answer b = ...
  45. 45先化简，再求值 (x2−3x−1−2)÷1x−1( \dfrac {x^{2}-3}{x-1}-2)÷ \dfrac {1}{x-1}，其中xx满足 x2−2x−3=0x^{2}-2x-3=0．
  46. 46Simplify x2−9y23x2+7xy−6y2\dfrac{x^2-9y^2}{3x^2+7xy-6y^2}.
  47. 47Express 23x−1−31+2x\dfrac{2}{3x-1} - \dfrac{3}{1+2x} as a single term, in its simplest form. Answer ___
  48. 48The variables x and y are connected by the equation y=2x+6x−7y = 2x + \dfrac{6}{x} - 7. Some corresponding values of x and y ar…
  49. 49Simplify 4aw2+16a35w\dfrac{4}{aw^2} + \dfrac{16a^3}{5w}. Answer ...
  50. 50下列运算正确的是( ) A. −a+ba+b=−1\dfrac{-a+b}{a+b}=-1 B. (−a3b)2=a6b2(-a^{3}b)^{2}=a^{6}b^{2} C. a+a=a2a+a=a^{2} D. a2⋅4a4=4a8a^{2}⋅4a^{4}=4a^{8}
  51. 51The nth term of a sequence is given by Tn=6n−53nT_n = \frac{6n-5}{3n}. Use the formula to find T7T_7, giving your answer as an…
  52. 52Solve the equation 2(x−5)=3x−12(x-5)=3x-1. Answer xx = ... Solve the inequality 7−4y>3(y+2)7-4y>3(y+2). Answer ... Given that 1p−12q=13r\dfrac{1}{p}-\dfrac{1}{2q}=\dfrac{1}{3r}…
  53. 53On Monday, Dev goes on a 3.6 km run. His average speed for the first 1.2 km is xx km/h. Simplify and write down an expr…
  54. 54计算： (1)(1+1x)⋅xx2−1(2)2x2+y2x+y−x2+2y2x+y(1)(1+\dfrac{1}{x})\cdot \dfrac{x}{{{x}^{2}}-1}(2)\dfrac{2{{x}^{2}}+{{y}^{2}}}{x+y}-\dfrac{{{x}^{2}}+2{{y}^{2}}}{x+y}…
  55. 55计算6a−3+a+33−a\dfrac{6}{a-3}+\dfrac{a+3}{3-a}的结果是()(\quad) A. 9−aa−3\dfrac{9-a}{a-3} B. 11 C. −1-1 D. 22
  56. 56先阅读，再回答问题： 如果x1x_{1}，x2x_{2}是关于xx的一元二次方程ax2+bx+c=0(a≠0)ax^{2}+bx+c=0(a\neq 0)的两个根，那么x1+x2x_{1}+x_{2}，x1x2x_{1}x_{2}与系数aa，bb…
  57. 57Simplify a−7/(ab6)+(1/a3)−2a^{-7}/(ab^{6}) + (1/a^{3})^{-2}.
  58. 58若xy=3 \dfrac {x}{y}=3，求(1+y2x2−y2)÷xx−y(1+ \dfrac {y^{2}}{x^{2}-y^{2}})÷ \dfrac {x}{x-y}的值．
  59. 59Write as a single fraction in its simplest form 32a−1−21−2a\frac{3}{2a-1} - \frac{2}{1-2a}. Answer ___
  60. 60计算： (1)x2x−5+255−x(2)1x−1−xx2−1(3)x+2x2−2x−x−1x2−4x+4(4)aa+1+a−1a2−1(5)x2x−1−x−1(6)x2−1x2−2x+1⋅x−1x2+x(1) \dfrac {x^{2}}{x-5}+ \dfrac {25}{5-x}(2) \dfrac {1}{x-1}- \dfrac {x}{x^{2}-1}(3) \dfrac {x+2}{x^{2}-2x}- \dfrac {x-1}{x^{2}-4x+4}(4) \dfrac {a}{a+1}+ \dfrac {a-1}{a^{2}-1}(5) \dfrac {x^{2}}{x-1}-x-1(6) \dfrac {x^{2}-1}{x^{2}-2x+1}⋅ \dfrac {x-1}{x^{2}+x}…