Junior two mathematics one yuan quadratic equation application problem. Seek detailed solution,. Thank you! A train goes on at constant speed. When it enters a 300 meter long tunnel, the knife completely passes through a kind of experience for 20 seconds. It also knows that a fixed equal vertical beam at the top of the tunnel irradiates the train for 10 seconds to find the length of the train

Let v be the uniform velocity
Then, the length of the car is L = 10V
300+L=20V
Two style tandem: vehicle length L = 300m

A self-employed household uses 50000 yuan to do business. It gains a certain profit in the first year. It is known that the 50000 yuan plus the profit in the first year will make a total profit of 2612.5 yuan in the second year, and the profit in the second year is 0.5 percentage point higher than that in the first year. Q: what is the profit margin in the first year?

Suppose the profit rate of the first year is X. (50000 + 50000x) × (x + 0.5%) = 2612.550000x + 250 + 50000x2 + 250x = 2612.550000x2 + 50250x-2362.5 = 0, X2 + 1.005x-0.04725 = 0, (x + 1.05) (x-0.045) = 0, then the solution is x = - 1.05 (not suitable, rounding off) or x = 4.5%. Answer: the profit rate of the first year is 4.5%

1. Given the circle C: (x-1) + (Y-2) = 2, the coordinates of P are (2, - 1). Through the point P, make the tangent line of circle C, and the tangent points are a, B. (1) find the tangent lines PA, P

The equation of standard circle should be 2 2 (x-1) + (Y-2) = 2, then the center of circle C (1,2) is connected with CP, then CP2 = (1-2) 2 + (2 + 1) 2 = 10, passing P as tangent PA, then pa2 = pc2-ac2 = 10-4 = 6 by Pythagorean theorem, then the root sign is 6

8. If the tangent points are m and N respectively, then the straight line Mn passes through the fixed point

After half a day's calculation, sure enough, the fixed point is the focus (0,1 / 2), which is the same as my judgment
Let the directrix of y = x2 be y = - 1 / 2, and the tangent coordinates be m (x0, x0 ^ 2), n (x1, X1 ^ 2)
Let P (m, - 1 / 2) be any point of the directrix, then the slope of the straight line PM is equal to the tangent slope of M 2x0,
So the PM equation is y + 1 / 2 = 2x0 (x-m), PM passes through point m, so x0 ^ 2 + 1 / 2 = 2x0 ^ 2-2x0m
That is, x0 ^ 2 = 2mx0 + 1 / 2 ①
Similarly, there is X1 ^ 2 = 2mx1 + 1 / 2,
① (2) x0 + X1 = 2m
From the two-point formula Mn: (y-x0 ^ 2) / (x-x0) = (x0 ^ 2-x1 ^ 2) / (x0-x1) = x0 + X1 = 2m
The Mn equation is y = x0 ^ 2 + 2m (x-x0) = 2mx0 + 1 / 2 + 2mx-2mx0 = 2mx + 1 / 2
So Mn over fixed point (0,1 / 2)

In rectangle ABCD, AB is equal to three centimeters and ad is equal to nine centimeters. Fold the rectangle so that point d coincides with point B. the crease is the area of triangle Abe?

From the folding feature, we know that: de = be, let AE = x cm
Then be = (9-x) cm
According to Pythagorean theorem, it is concluded that
3²+（9-x）²=x²
The solution is: x = 5
So s △ Abe = 1 / 2x4x3 = 6 (CM & # 178;)

If A1 = 1-1m, A2 = 1-1a1, A3 = 1-1a2 The value of a2013 is______ (expressed by an algebraic expression with m)

∵a1=1-1m，∴a2=1-1a1=1-mm−1=-1m−1，a3=1-1a2=1+（m-1）=m，a4=1-1m，… Every three data cycle, ∵ 2013 △ 3 = 671, ∵ a2013 = A3 = M

Let f be a focal point of the ellipse, p be on the y-axis, the line pf intersect the ellipse at M and N, the vector PM = T1 times the vector MF, and the vector PN = T2 times the vector NF
What is T1 + T2? (the elliptic equation is an ellipse with focus on the x-axis, and the answer is expressed by a and b)

First set the coordinates of each point, point P can be represented by a parameter, and then express the vector according to the known conditions, and then list the equations according to the vector equality. The last equation will appear C, X1 + X2, x1x2. Then you can solve the ellipse and straight line simultaneously, and get the equation about X1 + X2, x1x2, and bring it into the first equation. It seems a little troublesome, but you can try it

Decomposing the square-14 of factor 2x in the range of real numbers

Square difference formula: original formula = 2x square - (√ 14) square
=（2x+√14）（2x-√14）
Lemma: A & # 178; - B & # 178; = (a + b) (a-b)
Nanya middle school XXX

The sphere o is the inscribed sphere of the cube abcd-a1b1c1d1 with edge length of 1. The total area of the cone with the bottom of the circle cut by the sphere o on the plane acd1 is?
The bottom area of the cone is π / 6. What's the side area? The total area seems to be 2 π / 3

It is the center of the sphere and also the center of the cube. The distance between the circle and the plane is equal to the diagonal of the cube, that is, the distance between the circle and the plane is equal to the diagonal of the cube, that is, the radius of the sphere is equal to half of the edge length of the cube, that is, the area of the bottom of the cone

What is the law of sequence 1,3,7,13,21,31? How to write the general formula?

3-1=2
7-3=4
13-7=6
21-13=8
31-21=10
That is, a2-a1 = 2
a3-a2=4
……
an-a(n-1)=2(n-1)
Add
an-a1=2*n(n-1)/2=n²-n
a1=1
an=n²-n+1