 ### 第十七章：指数函数与对数函数 exponential function and logarithmic function

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

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



The quadratic equation kx + 2(k + a)x + (k + b) = 0 has equal roots. Express k in terms of a and b. Find the range of k where the equation kx + x + 2 = (x + 1) has two unequal real roots.

k < 1 The quadratic equation (p + 1)x + 2px + (p + 2) = 0 has real roots. Find the range of values of p. Prove that 2ax + (3a + 2c)x + 2c = 0 has distinct real root for all real value of x.



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

5


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



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



Find the range of the values of c for which 3x + 5x +c is always positive. If the quadratic function f (x) = (a – 2)x – 2ax + a + 1 is never positive, find the greatest value of a.

-2


Find the values of a for which ax + 3x + 4a is

(a) positive for all the real values of x,

(b) negative for all the real values of x.

(a) a > (b) a < Prove that (mx – 1)(x – 2) = m has two distinct real roots for x R.