Math topic中文

Quadratic equations

solving quadratic equations in one unknown by

  • factorisation
  • use of formula
  • completing the square for y=x2+px+qy = x^{2} + px + q
  • graphical method

Practice questions

  1. 1The curve CC has equation y=x2+x+3x−2y = \frac{x^2 + x + 3}{x - 2}. Without using a calculator, find the set of values that yy…
  2. 2The difference of the two roots of the equation 3x2+9x−k=03x^2 + 9x - k = 0 is 13 units, where kk is a constant. Find the value…
  3. 3在平面直角坐标系xOy中，已知A（2，-2），在y轴上确定点P，使△AOP为等腰三角形，则符合条件的点有 ( ) A. 2个 B. 3个 C. 4个 D. 5个
  4. 4方程∣xy∣+∣x+y∣=1\left|xy\right|+\left|x+y\right|=1的整数解的组数为()(\quad) A. 22 B. 44 C. 66 D. 88
  5. 5方程(x－1)(x＋2)＝0的两根分别是( ) A. x1＝－1，x2＝2 B. x1＝1，x2＝2 C. x1＝－1，x2＝－2 D. x1＝1，x2＝－2
  6. 6By completing the square or otherwise, express xx in terms of yy given that y=x2−4x+6y = x^2 - 4x + 6. Answer x = ___ or ___…
  7. 7Using 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 = .…
  8. 8关于xx的方程x2−bx+4=0x^{2}-bx+4=0有两个相等的正实数根，则bb的值为()(\quad) A. 44 B. −4-4 C. −4-4或44 D. 00
  9. 9已知两定点M(0,1)M(0,1)，N(1,2)N(1,2)，平面内一动点PP到MM的距离与PP到NN的距离之比为2 \sqrt {2}，直线y=kx−1y=kx-1与点PP的轨迹交于AA，BB两点． (…
  10. 10Sketch the graph of y=16−(x+3)2y = 16 - (x + 3)^2 on the axes below, indicating clearly the coordinates of the intercepts and the…
  11. 11已知(x2+y2)(x2+y2−5)=6(x ^{2} +y ^{2} )(x ^{2} +y ^{2} -5)=6，则x2+y2=x ^{2} +y ^{2} =___．
  12. 12用配方法解方程：x2+x−1=0x^{2}+x-1=0，配方后所得方程是()(\quad) A. (x−12)2=34(x-\dfrac{1}{2})^{2}=\dfrac{3}{4} B. (x+12)2=34(x+\dfrac{1}{2})^{2}=\dfrac{3}{4}…
  13. 13已知实数a,b,ca,b,c满足(a−b)2+b2+c2−8b−10c+41=0.(1){{(a-b)}^{2}}+{{b}^{2}}+{{c}^{2}}-8b-10c+41=0.(1)分别求a,b,ca,b,c的值； (2)如果aa、bb、cc是正数，且满足a+b+c=9a+b+c=9…
  14. 14A 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…
  15. 15若函数y=(k−1)x+k2−1y=(k-1)x+k^{2}-1是正比例函数，则kk的值是()() A. −1-1 B. 11 C. −1-1或11 D. 任意实数
  16. 16Simplify 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…
  17. 17关于xx的一元二次方程ax2+bx+1=0.(1)a{{x}^{2}}+bx+1=0.(1)当b=a+2b=a+2时，利用根的判别式判断方程根的情况； (2)若方程有两个相等的实数根，写出一组满足条件的aa，bb的值，并求此时方程的根.
  18. 18求过A(1,0)A(1,0)与B(0,1)B(0,1)两点，且在xx轴上截得的弦长等于66的圆的方程．
  19. 19解方程： x2+4x−4=0x ^ { 2 } +4x-4=0
  20. 20已知△ABC是等腰三角形，BC=8，AB，AC的长是关于x的一元二次方程x2﹣10x+k=0的两根，则（ ） A. k=16 B. k=25 C. k=﹣16或k=﹣25 D. k=16或k=25
  21. 21关于xx的方程x2+kx+3=0x^{2}+kx+3=0有一根为−1-1，则k=k= ___ ，另一根为 ___ .
  22. 22How many different real numbers xx satisfy the equation (x2−5)2=16(x^2 - 5)^2 = 16? A. 0 B. 1 C. 2 D. 4 E. 8
  23. 23若关于xx的一元二次方程(m−1)x2+2mx+m+3=0(m-1)x^{2}+2mx+m+3=0有两个不等的实根，则mm的取值范围()(\quad) A. m<32m< \dfrac {3}{2} B. m<32m< \dfrac {3}{2}且m≠1     m≠1\:\:\:\:\: …
  24. 24已知x1,x2\mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}是一元二次方程4kx2−4kx+k+1=0\mathbf{4}\mathbf{k}\mathbf{x}^{\mathbf{2}}\mathbf{-4}\mathbf{kx+k+1=0}…
  25. 25在平面直角坐标系中，点OO为坐标原点，直线y=kx+k(k>0)y=kx+k(k > 0)与xx轴交于点AA，与yy轴交于点BB，过点AA的抛物线y=x2+bx+cy=x^{2}+bx+c与xx轴交于另一点P(1)P(1)…
  26. 26用公式法解方程3x2+4=12x3x^{2}+4=12x，下列代入公式正确的是()(\quad) A. x=12±122−3×42x=\dfrac{12\pm\sqrt{12^{2}-3\times 4}}{2} B. x=−12±122−3×42x=\dfrac{-12\pm\sqrt{12^{2}-3\times 4}}{2}…
  27. 27关于xx的一元二次方程x2−kx+4=0x^{2}-kx+4=0的两个实数根分别是x1x_{1}、x2x_{2}，且满足x12+x22−2x1−2x2−7=0x_{1}^{2}+x_{2}^{2}-2x_{1}-2x_{2}-7=0，则kk的值为 ___.
  28. 28方程x2−6x+5=0x^{2}-6x+5=0的两根是()(\quad) A. 11和55 B. −1-1和55 C. 11和−5-5 D. −1-1和55
  29. 29In a class of 36 students, each student is either in a sports team or a performing arts club. Each student is also eithe…
  30. 30在平面直角坐标系中，如果点PP的横坐标与纵坐标相等，则称点PP为和谐点，例如：点P(1,1)P(1,1)、(−2,−2)(-2,-2)、(0.5,0.5)(0.5,0.5)…，都是和谐点，若二次函数y=ax2+7x+c(a≠0)y=ax^{2}+7x+c(a≠0)的图象…
  31. 31在△ABC\triangle ABC中，∠C=90°∠C=90°，tan⁡A=43\tan A=\dfrac{4}{3}，AB=20AB=20，则AC=AC=___.
  32. 32Solve 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}…
  33. 33已知等比数列{an}(a1≠a2)\{a_{n}\}(a_{1}\neq a_{2})的公比为qq，且a7a_{7}，a1a_{1}，a4a_{4}成等差数列，则q=()q=() A. 11或−32- 32 B. −32- 32 …
  34. 34二项式(3x3+1x)n(3 \sqrt[3]{x}+ \dfrac {1}{x})^{n}的展开式中的各项系数和为PP，所有二项式系数和为QQ，若P+Q=272P+Q=272，求展开式中的常数项．
  35. 35当mm为何值时，关于xx的一元二次方程(2m+1)x2+4mx+2m−3=0(2m+1)x^{2}+4mx+2m-3=0． (1)有两个不相等的实数根； (2)有两个相等的实数根； (3)没有实数根．
  36. 36The 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…
  37. 37A factory prints posters for corporate events. The prices of printing posters of different sizes are listed below. The f…
  38. 38在△ABC\triangle ABC中，设边aa，bb，cc所对的角为AA，BB，CC，且AA，BB，CC都不是直角，(bc−8)cos⁡A+accos⁡B=a2−b2(bc-8)\cos A+ac\cos B=a^{2}-b^{2}…
  39. 39如果关于xx的一元二次方程kx2−3x−1=0kx^{2}-3x-1=0有两个不相等的实根，那么kk的取值范围是 ___ ．
  40. 40记SnS_{n}为递增等比数列{an}\{a_{n}\}的前nn项和，若a1a2a3=8a_{1}a_{2}a_{3}=8，a4=a3+4a_{4}=a_{3}+4，则S10S_{10}的值为 ___ .
  41. 41Part of the graph of a quadratic function is shown below. The graph passes through the point A (1, 15) and it cuts the x…
  42. 42By expressing x2−10x−2x^2-10x-2 in the form (x−h)2+k(x-h)^2+k, where hh and kk are constants, write down the coordinates of the mi…
  43. 43Town 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…
  44. 44Factorise a3−2a2b−4a+8b.a^3 - 2a^2b - 4a + 8b. Solve 3x+29x=17x−3.\dfrac{3x+2}{9x} = \dfrac{1}{7x-3}.
  45. 45在△ABC\triangle ABC中，AB=122AB=12 \sqrt {2}，AC=13AC=13，cos⁡∠B=22\cos ∠B= \dfrac { \sqrt {2}}{2}，则BCBC边长为 ___ ．
  46. 46Factorise 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 = ...
  47. 47已知抛物线y=ax2y=ax^{2}经过点A(2,1).(1)A(2,1).(1)求此抛物线的表达式;(2);(2)写出抛物线上点AA关于yy轴的对称点BB的坐标;(3);(3)求△OAB\triangle OAB的面积;(4);(4)…
  48. 48用配方法解下列方程： (1)x2−6x+10=0(1){{x}^{2}}-6x+10=0； (2)2x2−7x+6=0.(2)2{{x}^{2}}-7x+6=0.
  49. 49Sketch the graph of y=−(x−2)2+4y=-(x-2)^2+4 on the axes below. Indicate clearly the coordinates of the points where the graph cro…
  50. 50已知函数f(x)=ex−e2x2,f′(x)f(x)=e^{x}-\dfrac{e}{2}x^{2},f'(x)为f(x)f(x)的导函数，若f′(a)=f(a)f'(a)=f(a)，则a=()a=(\quad) A. 00 B. −1-1 C. 22 D. 00…
  51. 51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)}．
  52. 52已知x=1x=1是方程x2−5ax+a2=0x^{2}-5ax+a^{2}=0的一个根，求代数式3a2−15a−73a^{2}-15a-7的值．
  53. 53Complete the table of values for y=2x+20x−15y = 2x + \dfrac{20}{x} - 15. | x | 1 | 2.5 | 5 | 10 | 12.5 | 15 | 20 | |---|---|---|-…
  54. 54解方程2(5x−1)2=3(5x−1)2(5x-1)^{2}=3(5x-1)的最适当的方法是()(\quad) A. 直接开平方法 B. 配方法 C. 公式法 D. 因式分解法
  55. 55若x2−3xy−4y2=0x^{2}-3xy-4y^{2}=0，则xy= \dfrac {x}{y}= ___ ．
  56. 56一元二次方程x2−2x+1=0x^{2}-2x+1=0的二次项、一次项系数分别为()(\quad) A. 11，22 B. 11，−2-2 C. x2x^{2}，−2-2 D. x2x^{2}，−2x-2x
  57. 57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．
  58. 58若不等式x2−(a+1)x+a⩽0x^{2}-(a+1)x+a\leqslant 0的解集是[−4,3][-4,3]的子集，则aa的取值范围是()() A. [−4,1][-4,1] B. [−4,3][-4,3] C. [1,3][1,3] D. [−1,3][-1,3]…
  59. 59函数f(x)=xax+b(a,bf(x)=\dfrac{x}{ax+b}(a,b为常数))满足：点(2,1)(2,1)在f(x)f(x)的图象上，方程f(x)=xf(x)=x有唯一解． (1)求f(x)f(x)的解析式； (2)判断f(x)f(x)在(−2,+∞)(-2,+∞)…
  60. 60解方程：x2−3x−4=0x^{2}-3x-4=0．