Math topic中文

Fractional to quadratic

solving fractional equations that can be reduced to quadratic equations such as: 6x+4=x+3\dfrac{6}{x + 4} = x + 3; 1x−2+2x−3=5\quad \dfrac{1}{x - 2} + \dfrac{2}{x - 3} = 5

Practice questions

  1. 1Simplify 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…
  2. 2A factory prints posters for corporate events. The prices of printing posters of different sizes are listed below. The f…
  3. 3Town 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…
  4. 4Factorise a3−2a2b−4a+8b.a^3 - 2a^2b - 4a + 8b. Solve 3x+29x=17x−3.\dfrac{3x+2}{9x} = \dfrac{1}{7x-3}.
  5. 5下列分式中，当xx为何值时，分式的值为零？ (1)x2−25∣x∣+16(1)\dfrac{{{x}^{2}}-25}{|x|+16}； (2)∣x∣−4(x+4)(x−2)(2)\dfrac{|x|-4}{(x+4)(x-2)}．
  6. 6Solve the inequality 3x+13<2−5x4\frac{3x+1}{3} < \frac{2-5x}{4}. Answer ... It is given that a=2b−3cb+2ca = \frac{2b-3c}{b+2c}. Find aa w…
  7. 7Alan 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…
  8. 8John 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…
  9. 9Solve 5(2−x)2+1x−2=4\dfrac{5}{(2-x)^2}+\dfrac{1}{x-2}=4. Answer x = ...
  10. 10Complete the table of values for y=2x2−2x.Valuesaregiventoonedecimalplacewhereappropriate.Onthegridopposite,drawthegraphofy=\dfrac{2}{x^{2}}-2x. Values are given to one decimal place where appropriate. On the grid opposite, draw the graph of …
  11. 11The table of values for y=x+5x−7y = x + \dfrac{5}{x} - 7 is given below. | x | 0.5 | 1 | 1.5 | 2 | 3 | 4 | 5 | 6 | 7 | |---|--…
  12. 12A bag contains 150 chips. There are 60 blue chips, x red chips and y green chips in the bag. The probability of drawing…
  13. 13By using factorisation, solve (x - 3)(2x + 8) = -12. Answer: x = ... Solve the equation $$\frac{4}{2x-3} - \frac{3}{x+2} = 1.…
  14. 14Solve the equation ex−2−14(1e)x−2=5e^{x-2} - 14\left(\frac{1}{e}\right)^{x-2} = 5. Solve the equation log2(3x+1)−2log⁡22=log⁡2(2−x2)log_{2}(3x+1) - \dfrac{2}{\log_{\sqrt{2}}2} = \log_{2}(2 - x^{2})…
  15. 15A supermarket sold all of its toilet rolls at a price of $xx per pack in January. The revenue made from selling the packs of toilet rolls in January was $5940. Write down an expression in x, for the number of packets of toilet rolls sold in January. Answer ... In February, the supermarket ordered an additional 600 packs to the number sold in January and sold them at 50 cents more per pack. Write down an expression in x, for the total amount of money received in dollars, if all the packs of toilet rolls were sold in February. Answer $ ... The supermarket received $3870 more from the sales of toilet rolls in February as compared to January. Write down an equation in x to represent this information, and show that it reduces to $$…
  16. 16a=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 =…
  17. 17Write 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…
  18. 18Simplify 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…
  19. 19The variables x and y are connected by the equation y=32x+5x−7y = \frac{3}{2}x + \frac{5}{x} - 7. Some corresponding values of x…
  20. 20b=aa−1+2cb = \dfrac{a}{a-1} + \dfrac{2}{c} Find the value of bb when a=3a = 3 and c=4c = 4. Answer b = ... Rearrange the formula…
  21. 21Estate E and Estate W are 35 km apart. Jason cycles at an average speed of xx km/h from Estate E to Estate W. On his re…
  22. 22Solve 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}…
  23. 23对于两个不相等的实数aa、bb，我们规定minh{aminh\{a、b}b\}表示aa、bb中较小的数的一半，如minh{2,3}=1.minh\{2,3\}=1.那么方程minh{x,−x}=2+2xxminh\{x,-x\}= \dfrac{2+2x}{x}…
  24. 24Simplify 16x−4−x+13x2−5x+2\frac{1}{6x-4} - \frac{x+1}{3x^2-5x+2}. 2x2+x+z=y\sqrt{2x^2 + x + z} = y Evaluate yy when x=−12x = -\frac{1}{2} and z=52z = \frac{5}{2}…
  25. 25Chris cycled from A to C in a certain part of a country. He travelled for 50 km from A, at a constant speed of xx km/h…
  26. 26Complete the table of values for y=2x+9x−11y = 2x + \dfrac{9}{x} - 11. Values are given to 1 decimal place where appropriate. |…
  27. 27Express as a single fraction in its simplest form 27−3x−16−x.\frac{2}{7-3x} - \frac{1}{6-x}. It is given that v=3−5ww+2+9.v = \frac{3-5w}{w+2} + 9.…
  28. 28若分式x2−4x−2\dfrac{{{x}^{2}}-4}{x-2}的值为00，则x=()x=() A. x=2x=2 B. x=−2x=-2 C. x=±2x=\pm 2 D. x=0x=0
  29. 29Solve 2x−7=3(1−3x).\n\nAnswerx=...Solvetheinequalities2x-7=3(1-3x).\n\nAnswer x = ... Solve the inequalities 7x-1<13andand\frac{x+1}{2}\ge -2(x-2).\n\nAnswer ... Rea…
  30. 30下列方程中没有实数解的是()(\quad) A. 1−x+x−1=0\sqrt{1-x}+\sqrt{x-1}=0 B. x+1+2=0\sqrt{x+1}+2=0 C. 2y+6y=72y+\dfrac{6}{y}=7 D. x2−3x−6=0x^{2}-3x-6=0…
  31. 31Complete the table of values for y=2x+1x2−4y = 2x + \dfrac{1}{x^{2}} - 4. Values are given to two decimal places where appropria…
  32. 32已知集合A={a∣A=\{a|关于xx的方程 x+ax2−1=1 \dfrac{x+a}{x^{2}-1}=1有唯一实数解，a∈R}a∈R\}，用列举法表示集合A=A=___．
  33. 33Solve 2x+1−3=12x−5\dfrac{2}{x+1} - 3 = \dfrac{1}{2x-5}. Answer ...
  34. 34Simplify 5a36b+2ab4a2b\frac{5a^3}{6b} + \frac{2ab}{4a^2b}. Solve x8=50x\frac{x}{8} = \frac{50}{x}. Simplify 4v2−12pv+p−10v−5\frac{4v^2 - 1}{2pv + p - 10v - 5}…
  35. 35Solve the equation 5+2x=201+x5 + 2x = \dfrac{20}{1 + x}. Answer x = ... or ...