How to find the value of m when x 3 x 2 x m 3 0 have three different real roots? What is a double ro

1) Let f(x)=x 3-x 2-x+m 3, then f'(x) = 3x 2-2x-1, let f'(x)=0, get f'The two zeros of (x) are: -1 3 and 1; f''(x)=6x-2，f''(-1 3)=-4 0, so f(x) takes the maximum value m 3+5 27 at x=-1 3;f''(1)=4 0, so f(x) takes the minimum value m 3-1 at x=1. Obviously, when the maximum value of f(x) is greater than 0 and the minimum value is less than 0, f(x)=0 can have three different real roots, that is, m 3+5 27 0 and m 3-1 0, that is, when -5 (1 3) 3 m 1, f(x)=0 will have three different real roots.

2) Root problem. In this problem, f(x)=0 has at most a pair of double roots, and the double root occurs at the zero point of the first derivative, that is, the double root is -1 3 or 1, and the maximum or minimum value of f(x) is 0, that is, m=-5 (1 3) 3 or m=1. (3) The analytical method of f(x)=0 root is quite cumbersome, and it cannot be listed here, you can search for "unary cubic equation" on the Internet.

Let f(x)=x 3-x 2-x

g(x)=-m^3

x 3—x 2—x+m 3=0 has three different real roots, and f(x) and g(x) have three intersections.

min(f(x))x=-1/3

x=1max(f(x))=f(-1/3)=-1/27-1/9+1/3=5/27

min(f(x))=f(1)=1-1-1=-11<-m^3<5/27

The cubic root number 5 3 has three different solid roots when the above conditions are met.

When m = 1 or equal to - cubic root number 5 3 there is a double root.

Guo Dunyun: When m=1 the cubic equation has a double root, x=1,1, 1When m = the cubic root of 5 8, the cubic equation has three different real roots.

The original equation is x 3 - x 2 - x + 5 8 = 0

x 1 2) (x x 2 5 4) = 0 When (x 1 2) = 0, x = 1 2

When (x x 2 5 4) = 0.

The cubic equations given at x=1 4+( 21) 4, and x=1 4 ( 21) 40 m 1 have three different real roots. ）

Let the two roots be macro return x1 and x2

According to Vedading, the front hunger theory.

x1+x2=-b Basis Balance a(x1,x2 are opposite numbers to each other, and the sum is 0)(m+1) 3=0

m=-1

The equation x 2 -3x + m=0 has two equal real roots, and the sum = b 2 -4ac=9-4m=0, and the solution gets: m= 9 4 The original equation source skin is: x 2 -3x+ 9 4 =0 solve:

x 1 = x 2 = 3 2 So the answer is slippery staring: 9 4 , x 1 =x 2 =

x²-2x+m-3=0

Same as x -3x+2m 0 root is x1 and has:

x1²-2x1+m-3=0

x1²-3x1+2m＝0

Subtract the two equations to obtain x1=m+3 and substitute into the first equation to obtain :

m+3)^2-2(m+3)+m-3=0

m^2+5m=0

m=0 or 5

1) When m=0, the same root of the brother is 0+3=3

2) When m=-5, the square foot silver rent is the same as Lingzhaogen -5+3=-2

It is known that 3x + (m+4) x + (m+1) = 0 has two opposite roots, i.e., the sum of the two roots is 0

According to Vedd's theorem: the sum of the two roots is: -(m+4) 3

Therefore: -(m+4) 3=0, and the solution is: m=-4.

Habitat loss. Excessive deforestation accelerates the loss of Siberian tiger habitat and deprives Siberian tigers of prey such as roe deer, wild boar and muntjac, resulting in food shortages for Siberian tigers.

Let the same root be x1

then x1 2-2x1+m-3=x1 2-3x1+2m gives x1=m+3

Substitute the equation and shout good Sanling.

m+3)^2-2(m+3)+m-3=0

m=0, m=-5

Substituting the equations x 2-2x+m-3=0 and x 2-3x+2m=0 respectively.

When m=0, the same root is x=3

When m=-5, the same root Zheng excavates lead as x=-2

Since it's the same. x 2-2x+m-3=x 2-3x+2mx=m+3.

m+3)^2-2(m+3)+m-3=0

m^2+5m=0

m(m+5)=0

m1=0 The root of this hunger and Qing phase is x=3m2=-5 before the rotten, and the same root is x=-2

x^2-2x+m-3=0

x^2-3x+2m=0

Subtract the two formulas to give x-m-3=0

x=m+3 is the common root of the prün.

Substituting x 2-3x+2m=0.

m+3)²-3（m+3)+2m=0

m²+5m=0

m=0,m=-5

When m=0, x 2-2x + m-3=0 turns into filial lead x -2x-3=0x=3, x = -1

x 2-3x+2m=0 to x -3x=0

x=3,x=0

The common root is x=3

When m=-5, x 2-2x+m-3=0 becomes a missing trace x -2x-8=0x=4, x=-2

x 2-3x+2m=0 to x -3x-10=0x=5, x=-2

The common root is x=-2