If the equation x2-4x + k = 0 has a common root with the equation x2-x-2k = 0, then the value of K should be______ ．

If the equation x2-4x + k = 0 has a common root with the equation x2-x-2k = 0, then the value of K should be______ ．

Let the common root be α. Then the two sides of the equation x2-4x + k = 0 are α and 4 - α; the two sides of the equation x2-x-2k = 0 are α and 1 - α. The relationship between the root and the coefficient is as follows: & nbsp; α (4 − α) = K α (1 − α) = 2K; the solution is α = 0k = 0 or α = 7K = − 21. So when k = 0 or - 21, the two equations have a common root. So the answer is: 0 or - 21

We know the equation X2 - (K + 2) x + 2K = 0 about X. (1) prove that no matter K is any real value, the equation always has real roots. (2) if one side of the isosceles triangle ABC a = 1 and the lengths B and C of the other sides are exactly the two roots of the equation, find the perimeter of △ ABC

It is proved that: (1) the equation ∵ (?) = b2-4ac = (K + 2) 2-8k = (K-2) 2 ≥ 0, regardless of any real value of K, always has real roots. (2) there are two cases: (1) if B = C, the equation X2 - (K + 2) x + 2K = 0 has two equal real roots, ∵ (?) = b2-4ac = (K-2) 2 = 0

Test of solving fractional equation with letter coefficients

If x = a makes the simplest common denominator 0, then a is the increasing root of the original equation. If x = a makes the simplest common denominator not zero, then a is the root of the original equation. For example: X / (x + 1) = 2x / (3x + 3) + 1 multiply by 3 (x + 1) 3x = 2x + (3x + 3) 3x = 5x + 3x = - 3x =

Using the solution of integral equation of one variable, we should pay attention to: (1) (1) (2)

1. Whether the denominator is multiplied by each item 2. Pay attention to the sign 3. Whether the number of letters is correct

There is a quadratic trinomial for the letter equation a, the coefficient of which is - 2 / 3
If the constant term is - 5, then the quadratic trinomial is

a-2a²-5

If the function f (x) = x (x-C) 2 has a maximum at x = 2, then the value of constant C is

f(x)=x^3-2cx^2+c^2x
f'(x)=3x^2-4cx+c^2
There is an extreme value at x = 2
f'(2)=12-8c+c^2=0
C = 2 or C = 6
When C = 2
f'(x)=3x^2-8x+4=(3x-2)(x-2)
x x

The following pairs of functions are the same: A: y = (x ^ 2-1) / (x-1) and y = x + 1, B: y = lgx ^ 2 and y = 2lgx, C: y = 3 root sign [x ^ 3 (1-x)]
D: Y = under the root sign [x (x-1)] and (x under the root sign) * (x-1 under the root sign)

C selected by exclusion method
In option a, X ≠ 1 is not considered, in front of item B, X ≠ 0, in the back, X ＞ 0, in front of item D, X ≥ 1 or X ≤ 0, in the back, X ≥ 1

Judge whether the following functions are the same and explain the reason: 1: the absolute value of y = x, y = the square of root x, 2: y = lgx2, y = 2lgx

1. Different
The absolute value of y = x = {x, x > = 0
-x,x0

Why are the two functions f (x) = lgx 2 and G (x) = 2lgx different, y = 2x + 1 and x = 2Y + 1 the same

Same function: the definition and value range of function are the same, and the function relationship is the same
Function is to express a certain relationship, independent of letters, y = 2x + 1 (the independent variable is x) and x = 2Y + 1 (the independent variable is y) represent the same relationship, and the domain is the same
F (x) = lgx & sup2; ① is different from the definition field of function g (x) = 2lgx ②: the definition field in ① is a non-zero number, and the definition field in ② is a positive number

1. Given the function f (x) = x square, find f (x-1)
2. Given the function f (x-1) = x square, find f (x)
How to start with such questions in the future? Mainly tell me what the method is?

Replace with
Replace x with X-1
f(x-1)=(x-1)²
2. Replace x with x + 1
f(x+1-1)=f(x)=(x+1)²