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

Find the range of k where the equation kx+ x + 2 = (x + 1)
has two unequal real roots.
求k 的范围当方程式 kx+ x + 2 = (x + 1)
有两个相异的实根。
Answer:

The quadratic equation (p + 1)x + 2px + (p + 2) = 0 has real roots. Find the range of values of p.
一元二次方程式(p + 1)x + 2px + (p + 2) = 0 有实根. 求 p值的范围.
Answer:

[4]
Prove that 2ax + (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 + (3a + 2c)x + 2c = 0 都有相异的实根.
Find the value of k so that the line kx + y = 8 is the tangent of the curve x + xy = 4.
求k 的值使得直线 kx + y = 8 是曲线 x + xy = 4的切线.
Answer:
Prove that, for all real value of a, the equation x + 2ax + 2a
+ a + 1 = 0 has no real roots for x.
试证对所有的实数a, 方程式 x + 2ax + 2a
+ a + 1 = 0 的x都没有实根.
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.
求c 值的范围使得 3x + 5x +c 恒为正。
Answer:

If the quadratic function f (x) = (a – 2)x – 2ax + a + 1 is never positive, find the greatest value of a.
若f (x) = (a – 2)x – 2ax + a + 1 是非正二元一次函数, 求 a的最大值.
Answer:
Find the values of a for which ax + 3x + 4a is
求a 值使得 ax + 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:

(b) a <
Prove that (mx – 1)(x – 2) = m has two distinct real roots for x R.
试证在x R,(mx – 1)(x – 2) = m 有两个相异的实根。