Given the equation about X, (m-1) * X & sup2; - 2mx + M = 0, there are two different real roots x1, X2, & sup2; = 8, find M

Given the equation about X, (m-1) * X & sup2; - 2mx + M = 0, there are two different real roots x1, X2, & sup2; = 8, find M

First of all, the original equation has two different real roots, so the original equation is a quadratic equation
M-1 ≠ 0 and discriminant
△=(-2m)^2-4(m-1)m>0
The simultaneous solution of the system of inequalities is m > 0 and m ≠ 1
By Weida theorem
x1+x2=2m/(m-1)
x1x2= m/(m-1)
So from & sup2; = 8
²-4 x1x2=8
[2m/(m-1)]²-4[m/(m-1)]=8
By solving the equation (M / (m-1) = t can be changed to simplify the calculation)
M = 2 or 1 / 2
According to the empirical calculation, it is consistent with M > 0 and m ≠ 1
M = 2 or 1 / 2

Proof: the equation x & sup2; + 2mx + m - 3 = 0 of X must have two unequal real roots

It is proved that B ^ 2-4ac = 4m ^ 2-4m + 12 = (2m-1) ^ 2 + 11
∵(2m-1)^2>=0 ∴(2m-1)^2+11>0
The equation x & sup2; + 2mx + M-3 = 0 must have two unequal real roots

It is known that the equation x ^ 2-2mx + m + 2 = 0 has two real roots
Then the minimum value of the sum of squares of the two real roots is?

Because the equation has real roots, so (- 2m) ^ 2-4 (M + 2) 0 = = = > m (- 1 or m) 2. According to Weida's theorem, two squares s = 4 (m-1 / 4) ^ 2-17 / 4. From the value range of M, the minimum value of the sum of two squares s smin = 2. If the value range of M is not considered, it is easy to get the minimum value = - 17 / 4

If f (x) is larger than f (2-x), then the value range of X is?

F (x) is the monotone increasing on the interval of (0, + ∞), f (x) > F (2-x),
Then x > 0, 2-x > 0, and x > 2-x
The solution is 1

Is it wrong for vector to judge coplanar theorem?
There is a necessary and sufficient condition when using vector to judge four points coplanar: for any point O in space, let OP = xoa + yob + Zoc. X + y + Z = 1
When judging the necessity: let ACBP form a parallelogram, because o is any point in the space, and select o as the midpoint of AB, then OP = xoa + xob OC, where x can be any value, so 2x-1 is not necessarily equal to 1
Isn't the theorem wrong?
But it is written in the mathematics textbook of pep. This is a necessary and sufficient condition
Should we add the condition that O is not the midpoint of AB?

OP = xoa + yob + Zoc, (x + y + Z = 1). This is the coplanar theorem of vectors. You said that O is the midpoint of AB, Op = - OC
Xoa + yob OC = OP, that is, x + Y-1 = 1. Your misunderstanding is that O is the midpoint of AB, then x = y. so you will have the idea that 2x-1 is not necessarily equal to 1. Of course, some ideas are good after all. It proves that you are thinking. If you continue, you will be successful in mathematics. If there is anything you can't, you can continue to ask me

As shown in the figure, △ ABC, ab = AC, BD and CE are the midlines on the sides of AC and ab respectively, and BD and CE intersect at point O. try to explain od = OE
file://C :\Documents and Settings\Administrator\Local Settings\Temporary Internet Files\ Content.IE5 \5fnndhce-20100511220000-450132920 [1]. JPG is a graph with a reason, such as ∵ AB = CD CD = BC ∵ AB = BC (equivalent substitution)

In fact, both sides are symmetrical, if you want to say a reason is congruent triangle! Triangle abd congruent ace
as follows
Find an equal angle on both sides and wait for everything
AB = AC = 2ae = 2ad (midpoint)
Angle a = angle a
So it's all waiting, and the written answer like the landlord's will be considered by the landlord himself

It takes a detailed process to ask a few math questions (if you answer well, you will get extra points). Thank you very much ("^" means power "*" means multiplication sign "/" means division sign)
Special statement: how many calculations can you do,
1. If the road is m long and N trees are planted, then the following formula can be used: (1) n = 2m / 5 + 1 (2) n = 2 (M / 5 + 1) (3) M = 5N / 2 (4) M = 5 (n / 2-1) a (1) (3) B (1) (4) C (2) (3) d (2) (4)
2. If A-B = 2, B-C = - 3, C-D = 5, then (A-C) (B-D) / (A-D)=_______
3. Put 1,2,3 The 100 natural numbers, 100, are randomly divided into 50 groups with 2 numbers in each group. Now, record any value of the two numbers in each group as a and the other as B, and substitute them into the algebraic formula (1 / 2) {[(a-b) absolute value] + A + B} to calculate the result. After 50 groups of numbers are substituted, 50 values can be obtained, then the maximum sum of the 50 values is________ The minimum value is_________ .
4. No matter what the value of K is, when x = 1, the formula (2kx + a) / 3 = 2 + (x-bk) / 6 always holds, and the values of a and B can be obtained
5. Given x ^ 2-3x-6 = 0, find the value of x ^ 3-5X + 2012

The first question is choice D,
The second question is to choose - 1 / 2,
In the fourth question, after substituting x = 1, we can get a, = 13 / 2, B = 9,

It is known that O is the center of circumscribed circle of triangle ABC. If 3oa vector + 4ob vector + 5oC vector = 0, then the angle ACB is?
Write down your ideas

3oa vector + 4ob vector + 5oC vector = 0 indicates that these three vectors form a triangle. According to Pythagorean theorem, it is known that the circle angle of right triangle = half of the center angle of the same chord = 45 degrees
Thank you!

As shown in the figure, in △ ABC, ad is the angular bisector, ∠ B = 60 ° and ∠ C = 45 ° to find the degree of ∠ ADB and ∠ ADC

In △ abd, the angle bisector of ∠ BAC is ∠ B = 60 °, the angle bisector of ∠ C = 45 °, the angle bisector of ∠ BAC is ∠ BAC, the angle bisector of ∠ CAD = 12, the angle of ∠ BAC = 37.5 °, in △ abd, the angle of ∠ ADC = 180 - bad - ADB = 97.5 °

Solution equation: x + 1x + 2 & nbsp; + & nbsp; X + 6x + 7 & nbsp; = & nbsp; X + 2x + 3 + & nbsp; X + 5x + 6

The original equation is changed to 1-1x + 2 + 1-1x + 7 = 1-1x + 3 + 1-1x + 6, so 1x + 2 + 1x + 7 = 1x + 3 + 1x + 6, and 1x + 2-1x + 3 = 1x + 6-1x + 7, that is, 1 (x + 6) (x + 7) = & nbsp; 1 (x + 2) (x + 3), so (x + 6) (x + 7) = (x + 2) (x + 3). X = - 92. X = - 92 is the root of the equation