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Given the set M = {(x, y) | y ^ 2 = 2x}, n = {(x, y) | (x-a) ^ 2 + y ^ 2 = 9}, the necessary and sufficient conditions for finding m intersection n to be an empty set are given
A necessary and sufficient condition for M intersection n to be an empty set
(x-a) ^ 2 + 2x = 9 no solution!
Or M = {(x, y) | y ^ 2 = 2x} is an empty set
A necessary and sufficient condition for the equation AX ^ 2 + BX + C = 0 (a, B, C belong to R, and a is not equal to 0) to have a root of 1
1) If the equation has a root x = 1, then a + B + C = 0 can be obtained by substitution; 2) when a + B + C = 0, C = - A-B, the equation is transformed into ax ^ 2 + bx-a-b = 0, and decomposed into (x-1) (AX + A + B) = 0, so the equation has a root x = 1. In conclusion, the equation AX ^ 2 + BX + C = 0 has a root of 1 if and only if a + B + C = 0
On the real coefficient equation AX2 + BX + C = 0 of X
1. What are the necessary and sufficient conditions for a root of the equation to be greater than 2 and a root to be less than 2;
2. What are the necessary and sufficient conditions for having two positive roots;
3. There is a positive root and a root of 0
Since there are two
So the discriminant is greater than 0
b^2-4ac>0
The following conditions should be met at the same time
Weida's theorem
x1+x2=-b/a,x1x2=c/a
1. One root is greater than 2 and one is less than 2
x12
So x1-20
So (x1-2) (x2-2) 0, B / A
On the equation AX & # 178; + BX + C = 0 (a is not equal to 0, a B C belongs to R), write a necessary and sufficient condition for satisfying the following conditions respectively
(1) The equation has two negative roots (2) the equation has one positive root and the other is zero
The first one is B & # 178; - 4ac > 0. The two negative roots are that the sum of two roots is less than 0, and the product of two roots is greater than 0
The sum of the second B & # 178; - 4ac > 0 is greater than 0, and the product of the two is equal to 0
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