The answer to the first question is:-root number 2 < = a < = 9/4.
The answer to the second question is:
When -2 < a < 0, the solution set of inequality is R.
When a=0, the solution set of inequality is {x|x is not equal to 1}.
When a=-2, the solution set of inequality is {x|x is not equal to-1}.
When a > 0 or a
{x | x > a+ 1+ radical sign (a 2+2a) or x
The answer to the third question is: {2, -2}
The answer is as follows:
The first question uses the idea of complement set.
The answer is:-root number 2 < = a < = 9/4.
The antonym of "at least one non-negative real root" is "either no real root or all negative roots"
X 2-(2a- 1) x+a 2-2 = 0。
The discriminant of roots = (2a- 1) 2-4 (a 2-2) =-4a+9.
The relationship between root and coefficient: x 1, x2 is the two roots of the equation,
x 1+x2=2a- 1,x 1*x2=a^2-2
If there is no real root, the discriminant of the root is < 0, that is, -4a+9 < 0.
Found a > 9/4
"All negative roots" means the discriminant of roots > = 0 and x 1+x20.
Namely -4a+9 > = 0 and 2a- 1 < 0 and a 2-2 > 0.
Namely a < = 9/4 and a < 1/2.
And (a) radical number 2 or a.
Synthesize to get a
In this way, "either there are no real roots or they are all negative roots."
A > 9/4 or a
In this way, the complement of the solution in the equation is-root number 2 < = a < = 9/4.
The necessary and sufficient condition for at least one nonnegative real root is
-radical number 2 < = a < = 9/4
The second question: classified discussion of problem-solving ideas.
Equation x 2-2 (a+ 1) x+ 1 = 0 corresponds to inequality.
The discriminant of roots = 4 (a+ 1) 2-4 = 4a 2+8a.
(1) When 4a 2+8a < 0, the solution set of inequality is r.
That is, when -2 < a < 0, the solution set of inequality is r,
(2) When 4a 2+8a = 0,
That is, when a=0, the solution set of inequality is {x|x is not equal to 1}.
When a=-2, the solution set of inequality is {x|x is not equal to-1}.
(3) when 4a 2+8a > 0,
That is, when a > 0 or
(by using the root formula)
{x | x > a+ 1+ radical sign (a 2+2a) or x
finally
When -2 < a < 0, the solution set of inequality is R.
When a=0, the solution set of inequality is {x|x is not equal to 1}.
When a=-2, the solution set of inequality is {x|x is not equal to-1}.
When a > 0 or a
{x | x > a+ 1+ radical sign (a 2+2a) or x
The third question: it is not easy for me to draw with the idea of combining numbers and shapes.
By 0
The equation x 2+ax+2 = 1 has only one solution.
Find a=2 or a=-2.
The set of real number A is {2, -2}