4x-x ^ 5 in the form of square difference and factorization

4x-x ^ 5 in the form of square difference and factorization

4x-x^5
=x（4-x^4）
=x(2-x^2)(2+x^2)
=-x(x^2+2)(x-√2)(x+√2)

The parabola C1: y = x & sup2 + 2x and C2: y = - X & sup2 + a are known
If the line L is tangent to both C1 and C2
Let l be the common tangent of C1 and C2
When we ask what value a is, C1 and C2 have and only have one common tangent
Write the equation of the common tangent

Let the common tangent be y = KX + B
The tangent points to C1 and C2 are (x1, kx1 + b), (X2, kx2 + b) respectively
Then the tangent points satisfy their respective parabolic equations
kx1 + b = x1² +2 x1
kx2 + b = -x2² + a
Derivation of C1 C2 equation
y1' = 2x + 2
y2' = -2x
The slope of the tangent is the derivative at the tangent point, so
k = 2x1 + 2 = -2 x2
All in all
kx1 + b = x1² +2 x1
kx2 + b = -x2² + a
k = 2x1 + 2
k = -2 x2
Four equations, K, x1, X2, B, a five unknowns
Eliminate B first
k(x1 - x2) = x1² + x2² + 2x1 -a
And then eliminate x1, x2
k(k-1) = (k/2 -1)² + (k/2)² + k -2 - a
k² - k = k²/2 -1 - a
k²/2 - k + 1 + a = 0
Discriminant
△ = b² - 4ac = 1 - 4*(1/2)*(1+a)
If there is only one common tangent, then
1 - 2(1+a) = 0
a = -1/2
k²/2 - k + 1/2 = 0
(k-1)² = 0
k = 1
x1 = (k-2)/2 = -1/2
y1 = x1² + 2 x1 = 1/4 - 1 = -3/4
x2 = -k/2 = -1/2
y2 = -x2² - 1/2 = -1/4 - 1/2 = -3/4
y1 = k*x1 + b
-3/4 = -1/2 + b
b = -1/4
Common tangent equation: y = x - 1 / 4
----------------------
Inspection:
For C1: y = x & sup2; + 2x
(- 1 / 2, - 3 / 4) on the curve
At this point, the tangent slope is k = 2x1 + 2 = 1
(- 1 / 2, - 3 / 4) on y = x - 1 / 4
For C1: y = - X & sup2; - 1 / 2
(- 1 / 2, - 3 / 4) on the curve
At this point, the tangent slope is k = - 2 x2 = 1
(- 1 / 2, - 3 / 4) on y = x - 1 / 4
The drawing shows that: under the condition of the title requirements, C1 and C2 are also tangent at (- 1 / 2, - 3 / 4)
Therefore, C1 and C2 equations can be directly established simultaneously
y =x² +2x
y = -x² +a
Find the value of a when there is only one intersection point between them
x² +2x = -x² +a
2x² +2x - a = 0
Using discriminant rule
a = -1/2
But this idea is not strict, because even if C1 and C2 have no intersection point, there may be a unique common tangent line. So I think strictly speaking, we still have to do the same trouble as above

All the concepts of junior high school physics
No problem sets

Outline of basic concepts of physics in junior middle school
1、 Measurement
1. Length L: main unit: meter; measuring tool: scale; when measuring, it should be estimated to the next digit of the minimum scale; the unit of light year is length unit
2. Time t: main unit: second; measuring tool: clock; stopwatch in laboratory. 1 hour = 3600 seconds, 1 second = 1000 milliseconds
3. Mass m: the amount of material contained in the object is called mass. Main unit: kilogram; measuring tool: scale; pallet balance for laboratory
2、 Mechanical movement
1. Mechanical movement: movement in which the position of an object changes
Reference: to judge the motion of an object, another object must be selected as the standard
2. Uniform linear motion:
① There are two ways to compare the speed of movement: A to compare the distance passed in the same time. B to compare the time required to pass the same distance
② Formula: 1 m / S = 3.6 km / h
3、 Strength
1. Force F: force is the action of objects on objects. The action of forces between objects is always mutual
Unit of force: Newton (n). Instrument for measuring force: dynamometer. Spring scale is used in laboratory
The effect of force: to deform or change the state of motion of an object
The change of object motion state refers to the change of object velocity or motion direction
The three elements of force: the size, direction and action point of force are called the three elements of force
The diagram of force should be scaled; the diagram of force should not be scaled
Gravity g: the force exerted on an object due to the attraction of the earth
Relationship between gravity and mass: g = mg m = g / g
G = 9.8 N / kg. Pronunciation: 9.8 N / kg, which means that the weight of an object is 1 kg and the gravity is 9.8 n
Center of gravity: the point of action of gravity is called the center of gravity of an object. The center of gravity of a regular object is at the geometric center of the object
4. Balance condition of two forces: acting on the same object; equal magnitude and opposite direction of two forces; acting on a straight line
Under the balance of the two forces, a body can be stationary or move in a straight line at a constant speed
The equilibrium state of a body means that the body is in a state of static or uniform linear motion. The resultant force of the external force on the body in equilibrium state is zero
5. The combination of two forces on the same line: the direction is the same: the resultant force F = F1 + F2; the resultant force direction is the same as that of F1 and F2;
The direction is opposite: the resultant force F = f 1 - F 2, the direction of resultant force is the same as that of large force
Under the same conditions, the rolling friction is much smaller than the sliding friction
Sliding friction is related to normal pressure, material properties and roughness of contact surface
7. Newton's first law is also known as the law of inertia. Its content is: all objects always keep static or uniform linear motion state when they are not affected by external force. Inertia: objects have the property of keeping original static or uniform linear motion state, which is called inertia
4、 Density
1. Density ρ: the mass per unit volume of a substance. Density is a property of a substance
Formula: M = ρ V, international unit: kggm3, common unit: g / cm3,
The relationship is: 1g / cm3 = 1 × 103kg / m3; ρ water = 1 × 103kg / m3; ρ water = 1 × 103kg / m3;
Reading: 103 kg per cubic meter, which means the mass of 1 cubic meter of water is 103 kg
2. Density measurement: mass measurement with pallet balance and volume measurement of solid or liquid with measuring cylinder
Area unit conversion:
1cm2 = 1 × 10-4m 2,
1 mm2 = 1 × 10-6 m2
5、 Pressure
Pressure P: the pressure on the unit area of an object is called pressure
Pressure F: force acting vertically on the surface of an object, unit: n
The effect of pressure is expressed by pressure, which is related to pressure and area
Pressure unit: n / m2; special name: Pascal (PA)
Formula: F = PS [S: area under force, common part of two objects in contact; unit: M2]
Methods to change the pressure: ① reduce the pressure or increase the stress area, can reduce the pressure; ② increase the pressure or reduce the stress area, can increase the pressure
2. Internal pressure of liquid: [measurement of internal pressure of liquid: use liquid manometer (U-tube manometer)]
Causes: due to the gravity of the liquid, pressure is produced on the bottom of the container; due to the fluidity of the liquid, pressure is produced on the wall of the container
Rules: ① at the same depth, the pressure in all directions is equal; ② the greater the depth, the greater the pressure; ③ different liquids at the same depth, the greater the liquid density, the greater the pressure
The formula: P = ρ GH H: unit: m; ρ: kg / m3; g = 9.8 N / kg
3. Atmospheric pressure: the atmospheric pressure is produced by the action of gravity. It is the madeburg hemispheric experiment that proves that atmospheric pressure exists and is very large. It is Torricelli (Italian scientist) who determines the value of atmospheric pressure. After the Torricelli tube is tilted, the height of the mercury column remains unchanged and the length becomes longer
1 standard atmospheric pressure = 76cm mercury column height = 1.01 × 105Pa = 10.336m water column height
Instruments for measuring atmospheric pressure: Barometer (mercury barometer, box barometer)
Atmospheric pressure changes with altitude: the higher the altitude, the lower the atmospheric pressure, that is, the higher the altitude, the lower the boiling point
6、 Buoyancy
1. Buoyancy and its cause: an object immersed in liquid (or gas) is supported upward by liquid (or gas), which is called buoyancy. Direction: vertical upward; cause: pressure difference between liquid and object
2. Archimedes principle: objects immersed in liquid are subject to upward buoyancy, and the buoyancy is equal to the gravity of objects expelling liquid
That is, f floating = g liquid discharge = ρ liquid GV discharge
3. Buoyancy calculation formula: F floating = G-T = ρ liquid GV discharge = f upper and lower pressure difference
4. When the object floats: F floats = g and ρ g and ρ 2F

1,11,101,1001,10001…… The general term formula of

When n > = 2, the general term is 1 + 10 ^ {n-1}; when n = 1, it is 1

It is known that the parabola C1 C2 is symmetric with respect to the Y axis, and the parabola c1c3 is symmetric with respect to the X axis. If the analytical expression of the parabola C2 is y = 3 / 4 (x + 2) - 1, find the solution of C3

The analytic expression of C2 is y = 3 / 4 * (x + 2) &# 178; - 1
The parabola C1 C2 is symmetric about the Y axis
So the analytic expression of C1 is y = 3 / 4 * (X-2) &# 178; - 1
The parabola c1c3 is symmetric about the x-axis
So the analytic expression of C3 is y = - 3 / 4 * (X-2) &# 178; + 1

A cylinder of silver is 38mm in diameter and 3mm in height. How many grams does the cylinder weigh?

0.019 ^ 2 * 3.14 * 0.003 * density of silver

May I ask 26 51 86 21 56 81
May I ask 26 51 86 21 56 81

It should be 26, which is the sum of the first of the first number and the second of the second number, such as 26 51

Find the original function f '(LNX) = 1 + LNX
F '(LNX) = 1 + LNX to find f (x)
I'll do it
lnx=t f'(t)=1+t f(t)=t+t^2/2+c f(x)=x+x^2/2+c
f(lnx)=lnx+ln(x)^2/2
f'(lnx)=1/x+lnx*(1/x)1+lnx
What's wrong?

f'(lnx)=1+lnx
That is, f '(x) = 1 + X
So f (x) = X-1

How many cubic meters of air does it take to produce 1kg of oxygen?

The oxygen density is 1.429
Oxygen volume = mass / density = 1000 / 1.429
The volume fraction of oxygen in the air is 21%
Air volume = (1000 / 1.429) / 21% = 6956 liters

Define the new operation AB as a natural number, specify a * b = (a + b) △ 2, and find the value of 3 * (4 * 6)

4*6
That is, a = 4, B = 6
4*6=(4+6)÷2=5
So the original formula = 3 * 5 = (3 + 5) △ 2 = 4