Method of preparation: 1) Find the minimum value of 2x ^ 2-7x-9, and determine the value of X at this time 2) Find the maximum value of - 3x ^ 2 + 5x + 4, and find the value of X at this time

Method of preparation: 1) Find the minimum value of 2x ^ 2-7x-9, and determine the value of X at this time 2) Find the maximum value of - 3x ^ 2 + 5x + 4, and find the value of X at this time

(1)2x^2-7x-9
=2(x²-7/2x)-9
=2(x-7/4)²-2×49/16-9
=2(x-7/4)²-121/16
When x = 7 / 4, the minimum value is - 121 / 16
(2)-3x^2+5x+4
=-3(x²-5/3x)+4
=-3(x-5/6)²+3×25/36+4
=-3(x-5/6)²+53/12
When x = 5 / 6, the maximum value is 53 / 12
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1. An amusement park invests 1.5 million yuan to introduce a large amusement facility. If the maintenance and insurance expenses are not included, it is estimated that the monthly income will be 330000 yuan. After the opening of the amusement facility, the accumulated maintenance and insurance expenses from the first month to the x month are y (unit: 10000 yuan), and y = ax ^ 2 + BX. The net income of the amusement park is g (unit: 10000 yuan), G is also a function of X
1) If the maintenance cost is 20000 yuan in the first month and 40000 yuan in the second month, find the function of Y with respect to X
2) Finding the functional relation of pure income g with respect to X
3) A few months after the opening of the amusement park, the net income of the amusement park will be the largest? Can the investment be recovered after a few months?
2. Assuming that the tariff rate of imported type a vehicles (hereinafter referred to as type a vehicles) is 100% in 2001 and 25% in 2006, the price of each type a vehicle in 2001 is 640000 yuan (including 320000 yuan tariff)
1) It is known that the price of B-type domestic cars (referred to as B-type cars) with similar performance to A-type cars is 460000 yuan in 2001. If the price of A-type cars is only affected by tariff reduction, in order to ensure that the price of B-type cars is 90% of that of A-type cars in 2006, the price of B-type cars should be reduced year by year. How many million yuan is the average annual decrease?
2) Someone invested 300000 yuan in 2004 and plans to buy a type B car with the investment and the return on investment by 2006. Assuming that the annual return on investment is the same, the return in the first year is included in the investment in the second year, and the minimum annual return is calculated (reference data: root 3.2 is about 1.79, root 1.2 is about 1.1)
3. A frame should be provided for a photo with a length of 29cm and a width of 22cm. The width of the four sides of the frame should be equal, and the area of the frame should be one fourth of the photo area. How many cm should the width of the side of the frame be?

1.
1) By substituting (1,2) (2,2 + 4) (that is, the accumulated maintenance cost in January is 20000 yuan, and the accumulated maintenance cost in February is 60000 yuan) into y = ax ^ 2 + BX, we can get a = 1, B = 1
y=x^2+x
2）g=33x-y-150=33x-（x^2+x）-150=-x^2+32x-150
3) G = - (x ^ 2-32x + 256) + 106 = - (x-16) ^ 2 + 106, so after 16 months of opening, the net income of the amusement park is 1.06 million yuan,
To recover the investment, if G > 0, X must be 6, that is to say, the investment will be recovered after 6 months
two
1) The actual price of model a car in 2001 was 320000 yuan, so the price of model a car in 2006 was 32 * (1 + 0.25) = 40 (including tariff)
Then the price of B-type car is 40 * 0.9 = 36, so the price of B-type car has dropped by 100000 yuan, so it has dropped by 20000 yuan every year for five years from 2001 to 2006
2) Suppose the rate of return is x, 30 (1 + x) ^ 2 = 36, then we get x = 0.1 or - 2.1 (- 2.1 rounding off), so the minimum rate of return is 10% per year
three
Let the frame width be X
The total area is (29 + 2x) (22 + 2x) = 4x ^ 2 + 102x + 638,
The frame area is 4x ^ 2 + 102x + 638-29 * 22 = 4x ^ 2 + 102x
The photo area is 29 * 22 = 638
4x^2+102x=1/4*638
The solution of the equation needs the help of a calculator. I've lost it in school. You can calculate it yourself
Typing is not easy,

As shown in the figure, PA and Pb are tangent to ⊙ o at two points a and B respectively, make the diameter AC, and extend the intersection Pb at point d to connect OP and CB. (1) prove op ∥ CB; (2) if PA = 12, DB: DC = 2:1, find the radius of ⊙ o

(1) It is proved that connecting AB, ∵ PA and Pb are tangent to ⊙ o at two points a and B respectively, ∵ PA = Pb and ∵ apo = ∵ BPO. ∵ op ⊥ AB & nbsp; & nbsp; ①. ∵ AC is the diameter of ⊙ o, ∵ ab ⊥ CB & nbsp; & nbsp; & nbsp; ②. From ① and ②, Op ∥ CB. (2) ∵ from (1) we know op ∥ CB, ∥ PBOC = DBDC. And ∵ Pb = PA = 12, DBDC = 21 ∥ 12oc = 21. ∥ OC = 6

It is known that in △ ABC, the coordinates of vertex a are (1,4), the linear equation of bisector of ∠ ABC is x-2y = 0, and the linear equation of bisector of ∠ ACB is x + Y-1 = 0

The symmetry points of point a about x-2y = 0 and X + Y-1 = 0 are all on BC. The symmetry points of point a (1,4) about x-2y = 0 and X + Y-1 = 0 are a '(a, b) and a' ', respectively

Two basic questions about the angle between,
1. What is the angle between the line y = 1 / 2x + 2 and the line y = 3x + 7
2. Given that the angle between X + 3y-15 = 0 and y-3mx + 6 = 0 is π / 4, what is m equal to

The formula of the angle between L1 and L2 is tan α = (k2-k1) / (1 + K1 * K2) 1. From y = 1 / 2x + 2, K1 = 1 / 2, from y = 3x + 7, K2 = 3, so tan α = (3-1 / 2) / (1 + 1 / 2 * 3) = 5 / 2 / (5 / 2) = 1  α = 45 degrees, then the angle between y = 1 / 2x + 2 and y = 3x + 7 is 45 degrees

If the line X / A + Y / b = 1 passes through the point m (COS β, sin β), then______ -Fill in the serial number
①a²+b²≤1 ②a²+b²≥1
③1/a² + 1/b²≤1 ④1/a²+1/b²≥1

Because the straight line x a + y B = 1 passes through the point m (COS β, sin β), the,
∴cosβ a + sinβ b =1,
We also point m (COS β, sin β) on the unit circle X & # 178; + Y & # 178; = 1,
So the line X / A + Y / b = 1 and the unit circle X & # 178; + Y & # 178; = 1 have a common point,
The distance from the center of circle to the straight line 1 / √ (1 / A & # 178; + 1 / B & # 178;) ≤ 1,
∴ (1 ／a )²+(1 ／b )² ≥1,
(4)

In the line passing through point a (- 3,1), the equation of the line farthest from the origin is______ ．

Let the origin be o, then the straight line passes through point a (- 3,1) and is perpendicular to OA, and koa = - 13, the slope of the straight line is 3, and its equation is Y-1 = 3 (x + 3), that is, 3x-y + 10 = 0. So the answer is: 3x-y + 10 = 0

The formula of profit and loss problem
In the problem of profit and loss, if there is more than one distribution, how can we find out if one distribution is just enough?

Divide the remainder by the difference between the two distributions

Profit and loss formula

Profit and loss
(profit + loss) △ the difference between the two distributions = the number of shares participating in the distribution
(big profit - small profit) △ the difference between the two distributions = the number of shares participating in the distribution
(big loss - small loss) △ the difference between the two distributions = the number of shares participating in the distribution

The formula of profit and loss problem

Hello
[profit and loss formula]
(1) if there is surplus (surplus) at one time and insufficient (deficit) at one time, the formula can be used as follows: (surplus + deficit) / (the difference between the number allocated by each person at two times) = the number of people
(2) there is surplus (surplus) in both times. The formula can be used as follows: (big surplus - small surplus) / (the difference of the number of people allocated in two times) = the number of people
(3) two times are not enough (deficit), the formula can be used: (big deficit - small deficit) / (the difference between the number of people allocated twice) = the number of people
(4) one time is not enough (loss), and the other time is just finished. The formula can be used: loss △ the difference between the number of people allocated in two times = number of people
(5) if there is surplus (surplus) in one time and the other time is just finished, the formula can be used as follows: surplus △ (the difference between the number allocated by each person in two times) = the number of people