Course Content
第二章：一元二次方程式与二次函数 quadratic equation
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2.2 一元二次方程式的根的判别式 The Discriminant of Quadratic Equations
2.3 一元二次方程式的根与系数的关系 The Relations of Roots and Coefficients of Quadratic Equations
2.4 一元二次函数的图像及性质 Graph and Nature of Quadratic Equation
2.5 一元二次函数的极值 The Extreme Values of Quadratic Functions
第三章：多项式 Polynomier
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3.3 综合除法 Synthetic Division – 练习题
3.4 余式定理 Remainder Theorem – 练习题
3.7 解一元高次方程式 Solve Higher-order Equation in One Variable – 练习题
第四章：部分分式 Partial Fraction
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4.3 部分分式 Partial Fraction – 练习题
第五章：无理式 Irrational Expression
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5.1 根式&无理式 Radical & Irrational Expression – 练习题
5.8 无理方程式 Radical Equations [练习题]
第六章：角的形成及单位 Angles and Measurements
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6.1 Angle 角 – 练习题
第七章：三角函数 Trigonometric Functions
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7.1 任意角的三角函数 Trigonomeric Function Value of Any Angle [练习题]
7.2 特别角的三角函数 Trigonomeric Function Value of Special Angle [练习题]
7.3 三角函数的诱导公式 Induced Formula of Trigonometric Functions [练习题]
7.5 三角函数的图像 Graphs of Trigonometric Functions [练习题]
第八章：任意三角形的解法 Solution of Triangles
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8.1-正弦定律 Sine Rule [练习题]
8.2 余弦定律 Cosine Rule [练习题]
8.5 三角形的外接圆半径和内切圆半径 Radius of Circumscribe Circle and Inscribe Circle of Triangles [练习题]
第九章：三角恒等式 Trigonometric Identities
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9.1 基本三角关系式 Basic Trigonometric Identities
9.2 两角和与差的三角函数 Compound Angles
9.3 倍角及半角的三角函数 Trigonometric Functions of Multiple Angle and Half Angle
9.4 三角函数的积化和差 Product Formulae I
9.5 三角函数的和差化积 Product Formulae II
第十章：三角方程式 Trigonometric Equations
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10.3 解三角方程式 Solve Trigonometric Equations
10.4 三角函数的图像 Graphing of Trigonometric Functions
10.5 三角方程式的图解法 Graphical Solutions of Trigonometric Equations
第十一章：直角坐标系 rectangular coordinate system
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11.1 直角坐标系 Cartesian System Coordinate
11.2 斜率 Gradient
11.3 三角形的面积 Area of Triangles
11.4 多边形的面积 Area of Poligons
11.5 分比公式 Dividing Ratio Formula
第十二章：直线方程式 Linear Equations
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12.2 直线方程式 Linear Equations
12.3 两条直线的平行与垂直 Parallel and Perpendicular Between Two Lines
12.4 两条直线的夹角 Angle Between Two Lines
12.5 两条直线的交点 Point of Intersection of Two Lines
12.6 点到直线的距离 Distance from a Point to Line
第十三章： 方程组 Simultaneous Equations
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13.2 方程组 Simultaneous Equations
第十四章：不等式 inequality
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14.3 不等式的证明 Proof of Inequalities
14.5 一元高次不等式 Higher Degree Inequalities in One Variable
14.6 分式不等式 Rational Inequalities
14.7 无理不等式 Irrational Inequalities
14.8 含绝对值的不等式 Inequalities Involving Absolute Values
14.9 代数式的最大值和最小值 Rational Quadratic Functions
第十五章：二元一次不等式及线性规划 linear inequality in two variables and linear programming
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15.2 二元一次不等式组 Binary Linear Inequalities
15.3 线性规划 Linear Programming
第十六意：数列与级数 sequence and series
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16.2 等差数列 Arithmetic Progression
16.3 等比数列 GeometricProgression
16.4 无穷级数 Infinite Series
16.5 简易特殊数列的和 The Sum of Some Simple Progression
第十七章：指数函数与对数函数 exponential function and logarithmic function
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17.1 指数 Indices
17.2 对数 Logarithm
17.3 对数的换底公式 Change of Base of Logarithms
17.4 指数方程式 Exponential Equations
17.5 对数方程式 Logarithmic Equations
高中一 | 高级数学
About Lesson

2.2 The Discriminant of Quadratic Equations 一元二次方程式的根的判别式

 

[1]

The quadratic equation kx^2 + 2(k + a)x + (k + b) = 0 has equal roots. Express k in terms of a and b.

一元二次方程式kx^2 + 2(k + a)x + (k + b) = 0 有等根. 用a 和 b 表示k.

Answer:
k = \frac{a^2}{b-2a}
[2]

Find the range of k where the equation kx^2+ x + 2 = (x + 1)^2 has two unequal real roots.

求k 的范围当方程式 kx^2+ x + 2 = (x + 1)^2 有两个相异的实根。

Answer:
k < 1\frac{1}{4}
[3]

The quadratic equation (p + 1)x^2 + 2px + (p + 2) = 0 has real roots. Find the range of values of p.

一元二次方程式(p + 1)x^2 + 2px + (p + 2) = 0 有实根. 求 p值的范围.

Answer:
p \leq -\frac{2}{3}

[4]

Prove that 2ax^2 + (3a + 2c)x + 2c = 0 has distinct real root for all real value of x.

试证对所有的实数x，2ax^2 + (3a + 2c)x + 2c = 0 都有相异的实根.

 

[5]

Find the value of k so that the line kx + y = 8 is the tangent of the curve x^2 + xy = 4.

求k 的值使得直线 kx + y = 8 是曲线 x^2 + xy = 4的切线.

Answer:
5
[6]

Prove that, for all real value of a, the equation x^2 + 2ax + 2a^2 + a + 1 = 0 has no real roots for x.

试证对所有的实数a, 方程式 x^2 + 2ax + 2a^2 + a + 1 = 0 的x都没有实根.

 

[7]

7Show that the line x + y = q will intersect the curve x^2 – 2x + 2y^2 = 3 in two distinct point if q^2 < 2q + 5. 试证若q^2 < 2q + 5，直线x + y = q 与曲线x^2 – 2x + 2y^2 = 3 交于两个相异的点。

 

[8]

Find the range of the values of c for which 3x^2 + 5x +c is always positive.

求c 值的范围使得 3x^2 + 5x +c 恒为正。

Answer:
c > \frac{25}{12}
[9]

If the quadratic function f (x) = (a – 2)x^2 – 2ax + a + 1 is never positive, find the greatest value of a.

若f (x) = (a – 2)x^2 – 2ax + a + 1 是非正二元一次函数, 求 a的最大值.

Answer:
-2
[10]

Find the values of a for which ax^2 + 3x + 4a is
求a 值使得 ax^2 + 3x + 4a 为

(a) positive for all the real values of x,
正对所有的实数x，

(b) negative for all the real values of x.
负对所有的实数x

Answer:
(a) a > \frac{3}{4}

(b) a < -\frac{3}{4}

[11]

Prove that (mx – 1)(x – 2) = m has two distinct real roots for x \epsilon R.

试证在x \epsilon R，(mx – 1)(x – 2) = m 有两个相异的实根。

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