Factorization (1) a ^ 4-b ^ 4 (2) x ^ 2 + 4 (3) x ^ 2 + 2x + 5 (4) a ^ 2 + B ^ 2 + C ^ 2 + 2Ab in complex number range

Factorization (1) a ^ 4-b ^ 4 (2) x ^ 2 + 4 (3) x ^ 2 + 2x + 5 (4) a ^ 2 + B ^ 2 + C ^ 2 + 2Ab in complex number range

(1) a^4-b^4
=(a^2-b^2)(a^2+b^2)
=(a-b)(a+b)(a+bi)(a-bi)
(2) x^2+4
=(x+2i)(x-2i)
(3) x^2+2x+5
=(x^2+2x+1)+4
=(x+1)^2-(2i)^2
=x+1-2i)(x+1+2i)
(4) a^2+b^2+c^2+2ab
=(a^2+b^2+2ab)+c^2
=(a+b)^2-(ci)^2
=(a+b+ci)(a+b-ci)

X ^ 2 + 2x + 10 = 0,

(x+1)²=-9
∴ x+1=±3
Ψ x = - 1 + 3I or x = - 1-3i

Why can a quadratic equation of one variable with real coefficients use the root formula, but the complex coefficients can not use the root formula

Because the discriminant B ^ 2-4ac in the complex coefficient equation may be complex, we must square it when finding the root; and complex square root, at least for high school students, is a very troublesome thing. So it is very difficult to solve the complex coefficient equation with the root formula, rather than unable to solve
Moreover, for any quadratic equation, it must be transformed into the form of (x-z1) ^ 2 = Z2 (Z1, Z2 are complex numbers) by using the collocation method. Next, as long as we can square Z2, we can solve it - as long as we can square Z2 - this is a new way to write the root formula
Example: x ^ 2 - (4 + 2I) X-1 + I = 0
After the formulation, (x - (2 + I)) ^ 2 = 4 + 3I was obtained
So x = 2 + I + (4 + 3I) ^ 0.5
The key is -- (4 + 3I) ^ 0.5 is equal to what?
If you have learned the square root of trigonometric form of complex number, of course, you can spend a little time to find out the following result (for the sake of writing simply, use M instead of root 2)
(4 + 3I) ^ 0.5 = (3 + I) / m or - (3 + I) / m
So the final solution is x = 2 + I + (3 + I) / m or x = 2 + I - (3 + I) / m
What if we use the root formula?
The discriminant is (4 + 2I) ^ 2-4 (- 1 + I) = 16 + 12I = 4 (4 + 3I),
So x = (4 + 2I + (4 (4 + 3I)) ^ 0.5) / 2
=2+i+(4+3i)^0.5.
It's the same as before, no difference
But if you haven't learned the root of complex number, you will be stuck in 4 + 3I
In addition, because the square root of the complex number is automatically signed, you don't need to write addition and subtraction in the formula, just write the plus sign directly
this is it.

To solve an equation, the solution is complex
1.w+w^2+w^3+w^4+w^5=-1
2.(2+5w+2w^2)^6=729
3.(1-w)(1-w^2)(1-w^4)(1-w^8)=9
It is not necessary to solve the whole problem.
1 solved: multiply W-1 on both sides

2. [(2 + 5W + 2W ^ 2) / 3] ^ 6 = 1, so [(2 + 5W + 2W ^ 2) / 3] ^ 3 = 1 or - 1, because the cube roots of 1 and - 1 are three known numbers, we can get six formulas by substituting them
3. Multiply both sides by 1 + W

A mathematical problem of rational number
Do you know the difference between the following groups?
(1) Positive numbers and positive rational numbers;
(2) Negative numbers and negative rational numbers;
(3) Fractions and decimals

Yes
1: Positive numbers include positive rational numbers and positive irrational numbers
2: Negative numbers include negative rational numbers and negative irrational numbers
3: Fractions are rational numbers; decimals include rational and irrational numbers, and more

(1) What are the characteristics of each logarithm of 4 and - 4,3 and - 3? (2) what is the position of the two logarithms on the number axis?

Man. We just finished. Huh
From the sign part, the sign of each logarithm is opposite, but their number parts are the same
Two opposite numbers are on both sides of the origin, and the distance to the origin is equal
I don't know if you always have problems after study or preview. But if you have problems, it's OK. Although it should also be homework

If the equation AX-1 = 3x + B about X has infinite solutions, what is the value of A-B 1
Fast solution of linear equation with one variable

According to the meaning of the title
a/2=3,-1=b
We get a = 6, B = - 1
1 / A-B = 6-1 / (- 1) = 7
1 / 7 of (a-b)

May I ask whether plury of, mass (ES) of, a great deal of, a great deal of, a quantity of is followed by a countable noun or an uncountable noun respectively, or whether both can be followed? After that, what words are the predicate verbs consistent with? Is it the word before of (for example, plural form of plury) or the modified noun after of?

First of all, the predicate must be consistent with the noun after of
A great deal cannot be followed by a noun
Please of, mass of, a great deal of
A quantity of followed by uncountable

In the triangle ABC, angle B = 2, angle a, ab = 2BC, prove angle a = 30 degrees

The angle B is equal to twice the angle a, and the angle EBD is equal to the angle a, so ad = dB, and the high de bisects AB, so be = BC is obtained from ab = 2BC, triangle BDE and triangle CBD are congruent, so angle c is equal to angle DEB, and de is the height of AB, so angle DEB = 90 degrees, so angle c is equal to 90 degrees

Write the best phrase in the class in English
A: what is the. Student in our class? B: Thank you for your help
Write more. The one above is just like

A:Who is the kindest student in our class?
B:Li Mei is the kindest student in our class.
A:Who is the most friendly student in our class?
B: Liu Ling is the most friendly student in our class