It is proved that the equation x * 2 with x power = 1 has at least one positive root of Xiaoyu 1

It is proved that the equation x * 2 with x power = 1 has at least one positive root of Xiaoyu 1

The equation is x * 2 ^ x = 1,
X = 0 is obviously not a root, so 2 ^ x = 1 / X
Let f (x) = 2 ^ X-1 / X
Then f '(x) = 2 ^ x * LN2 + 1 / x ^ 2 > 0, so the function increases monotonically and has only one root at most
And f (1) = 1 > 0
f(0.5)=2^0.5-2

Prove that the equation ln (1 + e ^ x) = 2x has at least one root less than 1

Constructor f (x) = ln (1 + e ^ x) - 2x
f(1)=ln(1+e)-2

The general solution of the differential equation y '' + 2Y = SiNx

Characteristic equation:
r² + 2 = 0
r = ±√2i
y = C₁sin(√2x) + C₂cos(√2x)
Let P = asinx + bcosx
P '' = - asinx - bcosx, which is substituted into the equation
(- Asinx - Bcosx) + 2(Asinx + Bcosx) = sinx
{ - A + 2A = 1 => A = 1
{ - B + 2B = 0 => B = 0
Special p = SiNx
The equation is y = C &; sin (√ 2x) + C &; cos (√ 2x) + SiNx

(2x to the second power, y + 3xy to the second power) - (6x to the second power, y-3xy to the second power) =?

（2x²y+3xy²）-（6x²y-3xy²）
=(2x²y-6x²y)+(3xy²+3xy²)
=-4x²y+6xy²
=2xy(3y-2x)

Simplify first and then evaluate. - (square of X + square of Y) + (- 3xy + square of X + square of Y), where x = 1 and y = 2

-(square of X + square of Y) + (- 3xy + square of X + square of Y)
=-x²-y²-3xy+x²+y²
=-3xy
When x = 1, y = 2
The original formula = - 3xy = - 6

If x + 3Y = 5, find the value of x ^ 2 + 3xy + 15y
It's better to have a process

If x + 3Y = 5, then y = (5-x) / 3
Bring in the original formula and simplify it

X-3y = 5, find x ^ 2-3xy-15y

Original formula = x (x-3y) - 15y
=5x-15y
=5(x-3y)
=5*5
=25

It is known that 2x-3y-z = 0, x + 3y-14z = 0, x, y, Z are not all 0, so how much is [4x square - 5xy + Z square] divided by [XY + YZ + ZX]/

Solve the equations
｛2x-3y-z=0.(1)
｛x+3y-14z=0.(2)
(1) + (2) is: 3x-15z = 0, that is: x = 5Z, substituting (1) is y = 3Z
So: (4x & # 178; - 5xy + Z & # 178;) / (XY + YZ + ZX)
=(100-75+1)z²/[(15+3+5)z²]
=26/23

Given that 2x-3y-z = 0, x + 3y-14z = 0, and X, y, Z are not all 0, find the value of (4x ^ 2-5xy + Z ^ 2) / (XY + YZ + ZX)

The answer is: 26 / 23
You can bring in x = 5, y = 3, z = 1 to verify the known conditions, check the correctness, and bring in the required formula to get the answer

Given that 2x-3y-z = 0, x + 3y-14z = 0, and X, y, Z are not all 0, then (4x2-5xy + Z2) / (XY + YZ + ZX)

The calculation is as follows: we know that 2x-3y-z = 0, x + 3y-14z = 0, and X, y, Z are not all 01) z = 2x-3y brings in X + 3y-14z = 0, x + 3y-28x + 42y = 0, that is, 45y = 27x gets x = 5 / 3Y, z = 2x-3y = Y / 32) and brings the above formula into the calculation (4x & # 178; - 5xy + Z & # 178;) / (XY + YZ + ZX) = (100 / 9-25 /