The mixed solution of 5x ^ 2 = 4-2x equation is solved by quadratic equation of one variable

5x²+2x-4=0
Discriminant = 4 + 80 = 84 = 4 × 21
So x = (- 2 ± 2 √ 21) / 10 = (- 1 ± √ 21) / 5
X = (- 1 + √ 21) / 5 or (- 1 - √ 21) / 5
Using the root method

On the univariate quadratic equation 2x & sup2; - 5x-n = 0 of X, when n satisfies what condition, the equation has a root of 0

Guess from the title
When x = 0
Bring in
-n=0
n=0
So when n = 0, x = 0

If the equation x2-2ax + 2 + a = 0 about X has two unequal real roots, find the value range of a satisfying the following conditions respectively
One equation is greater than 2 and the other is less than 2

Constructor y = x & # 178; - 2aX + 2 + A
If one of the equations is greater than 2 and the other is less than 2, only f (2) 2 is needed

If the equation y = LNX ax of X has a real root, then the value range of real number a

y'=1/x-a=0 ,y"=-1/x^2

How many real roots does the equation LNX = ax (where a > 0) have?
In advanced mathematics, I know the result!

Let f (x) = LNX ax, f '(x) = (1 / x) - A, Let f' (x) = 0, then x = 1 / A
When 0

If the equation LNX = x ^ 3-2ex ^ 2 + ax about X has two unequal real roots, then the value range of real number a

According to the value range of X, the original equation can be rewritten as: LNX / x = x ^ 2-2ex + A. This is equivalent to solving the problem that f (x) = LNX / X and G (x) = x ^ 2-2ex + a have two intersections. Because f '(x) = (1-lnx) / x ^ 2, when E > x > 0, f (x) increases monotonically, when x > e, f (x) decreases monotonically. Fmax = f (E) = 1 / E, and G (x) pairs

The maximum distance between the point on the ellipse x ^ 2 / 4 + y ^ 2 = 1 and the line 2x-4y-5 = 0

The translation line 2x-4y-5 = 0 forms a line bundle parallel to the line 2x-4y-5 = 0. The distances between the two parallel lines tangent to the ellipse and the original line are the minimum distance and the maximum distance respectively
Let the parallel line be 2x-4y + C = 0, and the equations of ellipse and line are established simultaneously
x^2/4+y^2=1
x^2/4+[（2x+c）/4]^2=1
Sort out the equation
8x^2+4cx+c^2-16=0
Δ = 16C ^ 2-32 (C ^ 2-16) = 0, so C = ± 1
Then the equation of two lines parallel to the original line is 2x-4y ± 1 = 0
The distance between two parallel lines is
|Under the root sign (2 ^ 2 + 4 ^ 2) = | ± 1 + 5 | / 2 √ 5
So the maximum distance is 6 / 2 √ 5 = (3 √ 5) / 5
The minimum distance is 4 / 2 √ 5 = (2 √ 5) / 5
Maximum distance = (3 √ 5) / 5

Let p be any point on the ellipse X & # 178 / 5 + Y & # 178; = 1, then the maximum distance from P to the straight line 2x-3y + 8 = 0 is

Know the ellipse (x ^ 2 / 4) + y ^ 2 = 1 and the point (0,2). (1) find a point P on the ellipse to maximize the value of | PA | (2) find the maximum distance between the point on the ellipse and the line 2x-3y-6 = 0
(1)P(±2√5/3,-2/3)
(2) Maximum 11 √ 13 / 13
Minimum √ 13 / 13

1. Let x = 2cosa, y = Sina
So | PA | - # 178; = x # 178; + (Y-2) # 178; = 4cos # 178; a + (sina-2) # 178; = 4cos # 178; a + sin # 178; a-4sina + 4 = - 3sin # 178; a-4sina + 8
=-3 (Sina + 2 / 3) & #178; + 28 / 3, so when Sina = - 2 / 3, the maximum value is 28 / 3, so the maximum value of | PA | is 2 √ 21 / 3
Cosa = ± 5 / 3, so p (± 2 √ 5 / 3, - 2 / 3)
2. Let the equation of the line parallel to the line and tangent to the ellipse be 2x-3y + C = 0
So y = 2 / 3x + C / 3 is substituted into the elliptic equation to get 25X & # 178; + 16cx + 4C & # 178; - 36 = 0
From △ = 0, 256c & # 178; - 100 (4C & # 178; - 36) = 0, C = ± 5
So the maximum value of distance is | C + 6 | / √ (4 + 9), so the maximum value is 11 / √ 13 and the minimum value is 1 / √ 13
That is, the maximum value is 11 √ 13 / 13, and the minimum value is 13 / 13

Using the method of advanced mathematics, we can find a point on the ellipse x &# 178; + y &# 178; = 4 to make it the shortest distance to the straight line 2x + 3y-6 = 0?

If it is an ellipse, let x = 2cosa, y = Sina, a belong to 0 to 2 π,
We obtain 13 under the absolute value / root of D = (4cosa + 3sina-6) and 13 under the absolute value / root of (5sin (a + b) - 6) with the auxiliary angle formula
If D is the smallest, then 5sin (a + b) = 5, and the final result is 13 / 13 under the root

The mixed solution of 5x ^ 2 = 4-2x equation is solved by quadratic equation of one variable