a: B = C: D (BD is not equal to 0), then a: (a-b) = C (C-D), please explain the reason

Because a: B = C: D
So ad = BC
That is - ad = - BC (multiply by - 1 at the same time)
AC ad = AC BC (plus AC)
That is: a (C-D) = C (a-b)
So a: (a-b) = C: (C-D)

It is known that a multiplied by B multiplied by C is not equal to 0. Try to prove that at least one of a multiplied by C, B multiplied by D and multiplied by D takes a positive value and one takes a negative value
It's better to talk about the process in detail. Thank you!
It is known that a times b times c times d is not equal to 0. Try to prove that at least one of a times C, negative B times D, B times D, c times d is positive, and at least one of them is negative.

Because ABCD ≠ 0
So ABCD is not zero
So BD ≠ 0, - BD ≠ 0
And because BD + (- BD) = 0
So the discussion is classified
1、 BD > 0, then - bd0, then - BD > 0
So there's at least one positive and one negative

a: B = C: D (ABCD cannot be equal to zero) try to explain that (AB + CD) is the middle term of the proportion of (a * a + C * c) and (b * B + D * d)

Square of (AB + CD) = square of AB + square of CD + 2abcd. 1
(a * a + C * c) and (b * B + D * d) = the square of AB + the square of CD + the square of BC + the square of AD
And because a: B = C: D (ABCD cannot be equal to zero), BC = ad,
Formula 1-2 = (BC AD) square, because BC = ad. so (BC AD) square > = 0
So, (AB + CD) is the median of the proportions of (a * a + C * c) and (b * B + D * d)

If the four real roots of the equation x & # 178; - 5x + M = 0 and X & # 178; - 10x + n = 0 are properly arranged, an equal ratio sequence with the first term of 1 will be formed
m: What is the value of N?

Let 1 be the root of the equation x & # 178; - 5x + M = 0
Then the two equations X & # 178; - 5x + M = 0 are 1 and 4 respectively
Two of the equations X & # 178; - 10x + n = 0 can be: 2, 8
In this case, 1:2 = 4:8
m=4,n=16
m:n=1:4

The calculation of recursive equation is 45 * 6 + 55

45×6+55
=45+55+45×5
=100＋45×2×2+45
=280+45
=325

The distance from point a (2,3) to the line 3Y + 4y-3 = 0 is

It should be 3x + 4y-3 = 0. You have the wrong number. D = | 2 * 3 + 3 * 4-3 | / the square of 3 under the root sign + the square of 4 = 3. The test point is the distance from the point to the straight line

Nine and five four plus 19 and five four plus 199 and five four plus 1999 and five four plus five four is equal to?

It is equivalent to 4 / 5 * 5 + 9 + 19 + 199 + 1999, that is, 10 + 20 + 200 + 2000 = 2230

Power series expansion of arctanx

First, write out the expression of variable upper limit integral of arctanx (all in the book), and then expand the integrand function with power series, exchange the integral sign and summation sign to get the result
But note that the exchange of integral sign and summation sign is conditional, to ensure uniform convergence, you can refer to the relevant information

When a divided by B is equal to 2 / 3, what is the square of a-3mn + 2B divided by the square of 2A + ab-3b

When a divided by B is equal to 2 / 3, what is the square of a-3mn + 2B divided by the square of 2A + ab-3b
The square of a-3ab + the square of 2B divided by the square of 2A + the square of ab-3b
=[(a/b)²-3(a/b)+2]/[2(a/b)²+(a/b)-3]
=(4/9)/(-13/9)
=-4/13

If the value of any real number x, y = ax ^ 2 + BX + C (b > A) is a non negative real number, the minimum value of a + B + C / B-A is

The value of y = ax ^ 2 + BX + C is always a nonnegative real number. When a = 0, △ ≤ 0, the value of Y is always a nonnegative real number,
△=b^2-4ac≤0
Because I can't understand the structure of a + B + C / B-A, I can't answer you, (is it (a + B + C) / (B-A)? Or B + C / b? Or a + B + C / (B-A)?)

a: B = C: D (BD is not equal to 0), then a: (a-b) = C (C-D), please explain the reason