Solution equation: 2x ^ 4-9x ^ 3 + 14x ^ 2-9x + 2 = 0, more details

Solution equation: 2x ^ 4-9x ^ 3 + 14x ^ 2-9x + 2 = 0, more details

2X ^ 4-9x ^ 3 + 14x ^ 2-9x + 2 = 0 2x ^ 4-2x ^ 3-7x ^ 3 + 7x ^ 2-7x-2x + 2 = 0 (x-1) (2x ^ 3-7x ^ 2 + 7X-2) = 0 (x-1) (2x ^ 3-2x ^ 2-5x ^ 2 + 5x + 2x-2) = 0 (x-1) ^ 2 * (2x ^ 2-5x + 2) = 0 (x-1) ^ 2 * (X-2) * (2x-1) = 0, so the solution of the equation is x = 1, or x = 2, or x = 1 / 2
Solving equation √ (x ^ 2 + 14x + 33) + √ (x ^ 2 + 9x-22) = 5 √ (x + 11)
The original equation is:
√(X+11)*√(X+3)+√(X+11)*√(X-2)-5√(X+11)=0
√ (x + 11) = 0 or √ (x + 3) + √ (X-2) - 5 = 0
X = - 11 or
X+3-10√(X+3)=X-2,
-10√(X+3)=-5
4(X+3)=1
4X=-11
X = - 11 / 4
The root of the original equation is x = - 11
5 / 2x-8 = 1 (x = 15, x = - 5) 2 / 3x-1-10 / 6x + 1 = 2 / 4x + 1 test whether the numbers in brackets of the following questions are the solutions of the equation in front of him
2/3x-1-10/6x+1=2/4x+1 （x=1/4 x=1/6)
1.5/2x－8＝1（x=15 x=-5）
===>5/2 X = 1+8 = 9
===>X = 10 ÷ 5/2 = 3.6
So the answers in brackets are all wrong (x = 15, x = - 5)
2.2/3x-1-10/6x+1=2/4x+1
2/3x-10/6x-2/4x = 1 +1 -1 = 1
24/36x - 60/36x - 18/36x = 1
-54/36x = 1
x= 1 ÷ -54/36
x= -36/54 = - 2/3
It's not easy to answer because there are no brackets
f bjnp[gm[hm
It is best to have an example of the method of finding the characteristic root, characteristic equation and general term formula in the problem of sequence of numbers
A (n + 2) = PA (n + 1) + QaN, P, q are constants
(1) Generally, let a (n + 2) - MA (n + 1) = k [a (n + 1) - man],
Then M + k = P, MK = - Q
(2) Characteristic root method
The characteristic equation is Y & # 178; = py + Q (*)
Note: ① m n is (*) two
② M n can exchange positions, but its results may appear in two different sequence forms, but it can also calculate an, and there will be unexpected surprises, hehe
③ After exchanging the positions of M N, we can construct two groups of recurrence formulas of an and a (n + 1). At this time, you will find that this is a system of quadratic equations of two variables about an and a (n + 1). Then we can eliminate a (n + 1) and leave an. OK, an is solved
Example: A1 = 1, A2 = 1, a (n + 2) = 5A (n + 1) - 6An,
The characteristic equation is: Y & # 178; = 5y-6
So, M = 3, n = 2, or M = 2, n = 3
Thus, a (n + 2) - 3A (n + 1) = 2 [a (n + 1) - 3an] (1)
A（n+2）-2A（n+1）=3[A（n+1）-2An] (2)
So, a (n + 1) - 3A (n) = - 2 ^ n (3)
A（n+1）-2A（n）= - 3 ^ (n-1) (4)
The elimination of a (n + 1) is an,
An=- 3 ^ (n-1) +2 ^ n.
For example:
(1)An=2A(n-1)+3A(n-2),A1=1,A2=3
The characteristic root is X1 = - 1, X2 = 3
-c1+3c2=1
c1+9c2=3
We get C1 = 0, C2 = 1 / 3
So an = 3 ^ (n-1)
(2)An=2A(n-1)-A(n-2),A1=1,A2=3
The characteristic root is X1 = x2 = 1
c1+c2=A1=1
(c1+2c2)×1=A2=3
We get C1 = - 1, C2 = 2
So an = (2n-1) × 1 ^ (n-1) = 2N-1
How to test 6x + 1 = 4x-3 is what both sides of the equation do at the same time, and then merge the similar terms,
6x+1=4x-3
2x=-4
x=-2
test
Equation left = 6 * (- 2) + 1 = - 11
Right side of equation = 4 * (- 2) - 3 = - 11
Equation left = right
So x = - 2 is the solution of the equation
Simultaneously - 1 or simultaneously + 3.
6X = 4x-4 or 6x + 4 = 4x
4x-6x=4
x=-2
Who can use eigenvalue equation to find the formula of sequence?
The characteristic equation eigenvalue method is used to solve the general term formula of sequence of numbers 1: a (n + 1) = pan + Q, P, q are constants. (1) generally, let a (n + 1) - λ = P (an - λ), then λ = q / (1-p). (2) if the characteristic root method is used here, the characteristic equation is x = PX + Q, and its root is x = q / (1-p). Note: if the characteristic root method is used, the coefficient of λ should be
4X + 3 = 3x-1 and equation 8x + 3 = 6x+_________ Fill in a number
From the first equation: 4x = 3x-4, x = - 4
Substituting: - 32 + 3 = - 24 +?
-29=-24+?
?=-29-（-24）
?=-5
Solve the left equation and get x = - 4. Substitute it into the right equation and get the coefficient of - 5
-5
How to deduce the expression of spring's elastic potential energy
How to deduce the expression of spring's elastic potential energy E = (1 / 2) * KX ^ 2?
So why is the area of the image the elastic potential energy of the spring?
Suppose an object rises slowly from the ground to height h under the action of gravity
In a limited altitude, gravity can be regarded as a constant mg, which does not change with altitude
So the work done by gravity on an object is - MGH
Gravity belongs to conservative force, the work done by conservative force + potential energy of conservative force = constant
Therefore, the expression of gravitational potential energy is MGH
------------------------------
For an elastic system, the elastic restoring force F = - KX
(k is the coefficient of elastic recovery, X is the distance from the equilibrium position)
Unlike gravity, the elastic restoring force is not constant and varies with the displacement X
Therefore, this topic needs the foundation of calculus knowledge
When the distance from the equilibrium position is x, the restoring force is f = - KX, and the negative sign indicates that the direction of the restoring force points to the equilibrium position, where k is the elastic coefficient of restitution
From the equilibrium position to the x position, the work done by the restoring force is the definite integral of the product of restoring force and displacement from 0 to X
W = ∫ f * DX = ∫ - KX * DX = - KX ^ 2 / 2 (from 0 to x) = - KX ^ 2 / 2 - 0 = - KX ^ 2 / 2
The restoring force belongs to the internal force of elastic system, as well as the conservative force like gravity
Work done by conservative force = negative value of conservative potential energy change
The equilibrium position is taken as the zero reference point of potential energy
Elastic potential energy E = - w = KX ^ 2 / 2
Make the f --- x relation curve. Make a vertical line from the beginning and end of the line to the X axis
Then the two vertical lines, X axis and F -- X curve form a closed figure
The area of this figure is the work w done by the force F
Have you ever been in touch with the above passage in middle school? If not, admit it directly. For the theory that cannot be proved due to insufficient knowledge reserve, admit it directly for the time being, which is also a common learning method
For this topic,
Taking the elastic force F = - KX as the y-axis,
Take the expansion amount x as the X axis
The F -- X "curve" is a straight line passing through the coordinate origin
A triangle in the fourth quadrant is obtained by projecting from the start and end of the line to the x-axis
The area of the triangle is
S = bottom * height / 2 = (x-0) * KX / 2 = KX ^ 2 / 2
Because the direction of the force is opposite to the direction of the displacement (and also because it is below the X axis), the work done by F is a negative value of the area, i.e
W = -S = -kx^2/2
And the elastic potential energy is
E = -W = kx^2/2
----------------------------------------------------
Why is the area of the image the elastic potential energy of the spring?
The formula of elastic potential energy is a very "basic" physical formula in middle school, but its derivation process is not found in textbooks. The reason is that its derivation process is beyond the scope of middle school students' knowledge
The students with a strong thirst for knowledge always want to know the derivation process. But after giving the derivation process, because the knowledge base is not enough, so they can't understand it, and there will be all kinds of questions. When these questions can't be solved, I hope you don't worry, because your knowledge reserve is insufficient
Simply answer your question
The area under the curve of dependent variable F as a function of independent variable x is the work done by F. this is a mathematical conclusion
You can imagine that if f is a constant, then the work done by F after the displacement x-x0 is f * (x-x0). Now mathematicalize this conclusion! Still do the F-X function image. Then the image is a straight line parallel to the X axis. The distance between the straight line and the X axis is f. therefore, the work f (x-x0) corresponds to the area of a rectangle on the function image, The rectangle is formed by projecting from the start and end of F line to X axis
In the previous discussion, f is a constant. Therefore, the expression of the work done by F is very simple. When the function image is no longer parallel to the x-axis, the work done by F is equal to the integral of F with respect to X. The mathematical concept of "integral" has not been touched in middle school, so it will be difficult for you to understand it, The result of "integration" is still the area of the figure formed by the projection of the function curve to the x-axis
2x-3 = 4x-9 solve the equation!
2x-4x=-9+3
-2x=-6
X=3
4x-2x=9-3
2x=6
X=3
2x-3=4x-9
2x-4x=3-9
-2x=-6
X=3
Move item = 4x-2x = 9-3 2x = 6 x = 3, hope the building owner adopt.
4x-2x=9-3
2x=6
X=3
Derivation of complete elastic collision formula
By momentum conservation:
m1*v1+m2*v1=m1*u1+m2*u2
Energy conservation:
0.5m1*v1^2+0.5m2*v2^2=0.5m1*u1^2+0.5m2*u2^2
If it is not completely eliminated, a relation can be obtained
v1+u1=v2+u2
Does this formula have any theorem or physical meaning?
Just change the formula
v1-v2=u2-u1
The left side is the relative velocity of object 1 approaching object 2 before collision. The right side is the relative velocity of object 2 leaving object 1 after collision. Therefore, the physical meaning is that the approaching velocity equals the departing velocity