Math topic中文

Factorisation quadratic

factorisation of quadratic expressions ax2+bx+cax^{2} + bx + c

Practice questions

  1. 1The equation of a curve is y=x2e2xy = x^2 e^{2x}. Obtain an expression for dydx\dfrac{dy}{dx}. Find the range of values of xx…
  2. 2Factorise 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}.
  3. 3Using factorisation, solve 9x2+6x−8=09x^2 + 6x - 8 = 0. Answer x = ... or ... Hence solve 9(y−1)2+6y−14=09(y-1)^2 + 6y - 14 = 0. Answer y = .…
  4. 4Factorise completely. abc−3c−6+2ababc - 3c - 6 + 2ab Answer ... 80x4−580x^{4} - 5 Answer ...
  5. 52x2−5xy−12y2x2−16y2\displaystyle \frac{2x^2-5xy-12y^2}{x^2-16y^2} Simplify. Answer ...
  6. 6A 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…
  7. 7Factorise the following completely. 45b−18ab−2a2+5a45b - 18ab - 2a^2 + 5a 2p2−23p−162p^2 - \dfrac{2}{3}p - \dfrac{1}{6}
  8. 8将多项式mn2+2mn+mmn^{2}+2mn+m因式分解的结果是 ___ ．
  9. 9Express −x2+4x+12-x^2+4x+12 in the form −(x+p)(x−q)-(x+p)(x-q). Answer ... Sketch the graph of y=−x2+4x+12y=-x^2+4x+12 on the axes below. Indicat…
  10. 10已知不等式ax2−3x+2>0a{x}^{2}-3x+2>0的解集为{x∣x<1或x>b}.(1)\{x|x< 1或x>b\}.(1)求aa、bb的值； (2)解关于xx的不等式x2−b(a+c)x+4c>0.{x}^{2}-b(a+c)x+4c>0.
  11. 11Solve 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}…
  12. 12已知p(x)p(x)：x2−5x+6<0x^{2}-5x+6 < 0，则使p(x)p(x)为真命题的xx取值范围为 ___ ．
  13. 13Solve 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…
  14. 14Solve 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}}…
  15. 15Factorise 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 = ...
  16. 16Factorise 9x2+24xy+16y29x^2 + 24xy + 16y^2 completely. Answer ... Hence factorise 144a8−(9a8+24a4y+16y2)144a^8 - (9a^8 + 24a^4 y + 16y^2) completely. A…
  17. 17若x2−3xy−4y2=0x^{2}-3xy-4y^{2}=0，则xy= \dfrac {x}{y}= ___ ．
  18. 18Express 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…
  19. 19解方程：x2−3x−4=0x^{2}-3x-4=0．
  20. 20Express 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…
  21. 21Factorise 3x2−14x−53x^2-14x-5. Answer ... Hence, factorise completely 3(y+1)2−14y−193(y+1)^2-14y-19. Answer ...
  22. 22将多项式x2−2x−8x^{2}-2x-8分解因式，正确的是()(\quad) A. (x+2)(x−4)(x+2)(x-4) B. (x−2)(x−4)(x-2)(x-4) C. (x+2)(x+4)(x+2)(x+4) D. (x−2)(x+4)(x-2)(x+4)
  23. 23It is given that 2x4−kx3−2kx2+5x+62x^4 - kx^3 - 2kx^2 + 5x + 6 leaves a remainder of 77 when divided by (2x−k).(2x - k). Show that k3−5k+2=0k^3 - 5k + 2 = 0…
  24. 24Given x2−8xy+16y2=0x^2-8xy+16y^2=0, find the value of xy\dfrac{x}{y}. Answer ...
  25. 25因式分解． （1）2a3b－8ab3 （2）3a2－2ab－8b2
  26. 26Solve 5(2−x)2+1x−2=4\dfrac{5}{(2-x)^2}+\dfrac{1}{x-2}=4. Answer x = ...
  27. 27The seating capacity of an auditorium is 300. The seats are categorised as 1, 2, 3 and 4. The costs of the tickets in th…
  28. 28Write as a single fraction in its simplest form. 3t2w⋅9t2w3\dfrac{3t^2}{w} \cdot \dfrac{9t^2}{w^3} Answer ----------------------…
  29. 29解方程： (1)(x+2)2−36=0(1)(x+2) ^{2} -36=0； (2)2x2−7x+6=0(2)2x ^{2} -7x+6=0．
  30. 30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)…
  31. 31A bag contains 150 chips. There are 60 blue chips, x red chips and y green chips in the bag. The probability of drawing…
  32. 32By using factorisation, solve (x - 3)(2x + 8) = -12. Answer: x = ... Solve the equation $$\frac{4}{2x-3} - \frac{3}{x+2} = 1.…
  33. 33Factorise -2x^2 + x + 3. Answer ... Factorise completely 8x^3 - 18xy^2. Answer ...
  34. 34Use factorisation to solve the equation. 2h2−11h−21=02h^2 - 11h - 21 = 0 Answer h = ... or ...
  35. 35已知2x2−3xy−2y2=02x^{2}-3xy-2y^{2}=0，且x>0x > 0，y>0y > 0，则x+yx−y \dfrac {x+y}{x-y}的值为 ___ ．
  36. 36Factorise completely 2x3−13x2+6x2x^3 - 13x^2 + 6x. Answer ...
  37. 37A supermarket sold all of its toilet rolls at a price of $xx per pack in January. The revenue made from selling the packs of toilet rolls in January was $5940. Write down an expression in x, for the number of packets of toilet rolls sold in January. Answer ... In February, the supermarket ordered an additional 600 packs to the number sold in January and sold them at 50 cents more per pack. Write down an expression in x, for the total amount of money received in dollars, if all the packs of toilet rolls were sold in February. Answer $ ... The supermarket received $3870 more from the sales of toilet rolls in February as compared to January. Write down an equation in x to represent this information, and show that it reduces to $$…
  38. 38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}…
  39. 39Factorise completely x4−y4x^4 - y^4. Answer ... Given that x2−2xy=1−y2x^2 - 2xy = 1 - y^2, express yy in terms of xx. Answer ... o…
  40. 40Simplify 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…
  41. 41Simplify x2−9y23x2+7xy−6y2\dfrac{x^2-9y^2}{3x^2+7xy-6y^2}.
  42. 42Solve the equation 13x2=3x\frac{1}{3}x^2 = 3x. Answer x = ... or ...
  43. 43方程x2+x−12=0x^{2}+x-12=0的两个根为()() A. x1=−2x_{1}=-2，x2=6x_{2}=6 B. x1=−6x_{1}=-6，x2=2x_{2}=2 C. x1=−3x_{1}=-3，x2=4x_{2}=4 D. x1=−4x_{1}=-4…
  44. 44Estate 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…
  45. 45不等式x2<4x+5{x}^{2}< 4x+5的解集为()(\quad) A. (−∞,−5)∪(1,+∞)(-{\rm∞},-5)∪(1,+{\rm∞}) B. (−∞,−1)∪(5,+∞)(-{\rm∞},-1)∪(5,+{\rm∞}) C. (−1,5)\left(-1,5\right)…
  46. 46Express x2+5x+4x^2 + 5x + 4 in the form (x+p)2+q(x + p)^2 + q. Answer (a) ... Sketch the graph of y=x2+5x+4y = x^2 + 5x + 4. Indicate clea…
  47. 47Factorise 6x2−x−26x^2 - x - 2. Answer ...
  48. 48x2−x−12x^{2}-x-12分解因式得___. .
  49. 49Simplify 9x2−412x2−x−6\dfrac{9x^2 - 4}{12x^2 - x - 6}. Answer ...
  50. 50Factorise completely 6x^2 + x - 2. Hence, factorise completely 6(3m - 1)^2 + 3m - 3.
  51. 51Given that a=b2+4ac2aa = \sqrt{\frac{b^2+4ac}{2a}}, find the values of bb when a=4a = 4 and c=−1c = -1. Answer b = ... Simplify (125p6q3)−1/3×(3p0q)2\left(\dfrac{125p^6}{q^3}\right)^{-1/3}\times\left(\dfrac{3p^0}{\sqrt{q}}\right)^2…
  52. 52It is given that y=5x−23x+1y=\dfrac{5x-2}{3x+1}. Find yy when x=7x=7. Answer yy = ... Express xx in terms of yy. Answer xx =…
  53. 53Factorise completely 9b2−6ab+a2−x29b^{2}-6ab+a^{2}-x^{2}. Simplify 52x2−7x−4−84−x\displaystyle\frac{5}{2x^{2}-7x-4}-\frac{8}{4-x}.
  54. 54Simplify 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}…
  55. 55Solve the inequality x−53−2x−12≤−1\frac{x-5}{3} - \frac{2x-1}{2} \le -1. Express as a single fraction in its simplest form 1x2−4−1x2−x−6\frac{1}{x^2-4} - \frac{1}{x^2 - x - 6}…
  56. 56Simplify 2x2+4xy−3x−6y2x2+xy−6y2\dfrac{2x^2+4xy-3x-6y}{2x^2+xy-6y^2}. Answer ...
  57. 57先化简再求值：(3x−1−x−1)÷x−2x2−2x+1( \dfrac {3}{x-1}-x-1)÷ \dfrac {x-2}{x^{2}-2x+1}，其中xx是方程x2=2xx^{2}=2x的根．
  58. 58在下列从左到右的变形中，不是因式分解的是()(\quad) A. x2−x=x(x−1)x^{2}-x=x(x-1) B. x2+3x−1=x(x+3)−1x^{2}+3x-1=x(x+3)-1 C. x2−y2=(x+y)(x−y)x^{2}-y^{2}=(x+y)(x-y) D. x2+2x+1=(x+1)2x^{2}+2x+1=(x+1)^{2}…
  59. 59化简求值：⑴(1+x+x21−x)÷1+x1−x,x=−3⑵(x+2x2−2x−x−1x2−4x+4)÷x−4x,x=−5⑴\left(1+x+ \dfrac{{x}^{2}}{1-x}\right)÷ \dfrac{1+x}{1-x},x=-3 ⑵\left( \dfrac{x+2}{{x}^{2}-2x}- \dfrac{x-1}{{x}^{2}-4x+4}\right)÷ \dfrac{x-4}{x},x=-5 …
  60. 60Chris 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…