Solution of equation (5 ^ x) * 10 ^ 3x = 8 ^ x The solution of equation 2 ^ (x + 1) = 3 ^ (2x + 1)

Solution of equation (5 ^ x) * 10 ^ 3x = 8 ^ x The solution of equation 2 ^ (x + 1) = 3 ^ (2x + 1)

(5^x)*10^3x=8^x
5^x*2^3x*5^3x=8^x
5^4x*8^x=8^x
5^4x=1
x=0
2^(x+1)=3^(2x+1)
2^x*2=3^2x*3
(2/9)^x=3/2
x=log(2/9)(3/2)
x=(log3-log2)/(log2-log9)

2 ^ (x ^ 2 + 3) = (1 / 4) ^ 7 / 2 solve exponential equation

2^(x^2+3)=(1/4)^7/2 =（2^-2）^7/2=2^-7
And x ^ 2 + 3 = - 7
The solution is x = ± √ 10

It is known that the three vertices of an equilateral triangle OAB are on the parabola y ^ 2 = 2x, where o is the origin of the coordinate, and let C be the circumscribed circle of the triangle OAB (point C is the center of the circle)

The slope of OA is Tan 30 ° = 1 / √ 3
The equation is y = x / √ 3, which is substituted into the parabolic equation y ^ 2 = 2x,
We get x = 0 or x = 6,
Substituting x, y = 2 √ 3
A(6,2√3),
The center of the circle is set as D (D, 0), d = 6 - (2 √ 3) Tan 30 ° = 4;
The radius is r, R & sup2; = | Da | & sup2; = (6-4) & sup2; + (2 √ 3-0) & sup2; = 16,
So the equation of circle is (x-4) & sup2; + Y & sup2; = 16

Take any 12 natural numbers and try to prove that the remainder of at least two natural numbers divided by 11 is the same

The remainder of natural number divided by 11 can only be {0,1,2 11 cases
So there must be less than two natural numbers out of 12 that have the same remainder divided by 11

In the polar coordinate system, O is the pole, and the polar coordinate of the center of the circle C with radius 2 is
In the polar coordinate system, O is the pole, the polar coordinate of the center of circle C with radius 2 is (2, π / 3), and the polar coordinate equation of circle C is obtained
To explain 】

If you are not used to it, you can convert all the coordinates to rectangular coordinates, and then convert them to polar coordinates. The center of the circle is (1, √ 3), and the radius is 2, so the equation is (x-1) ^ 2 + (Y - √ 3) ^ 2 = 4

Integral multiplication and division and factorization
1. It is known that the trilateral lengths a, B and C of △ ABC satisfy the relation - C & sup2; + A & sup2; + 2ab-2bc = 0, and try to explain that △ ABC is an isosceles triangle
2.1 × 2 × 3 × 4 + 1 = 25 = 5 & sup2;; 2 × 3 × 4 × 5 + 1 = 121 = 11 & sup2;; 3 × 4 × 5 × 6 + 1 = 361 = 19 & sup2;... According to the above rule, Xiaoqiang conjectures that the sum of the product of any four continuous positive integers and one must be a complete square. Is Xiaoqiang's conclusion correct? If it is correct, please prove it; if not, please explain the reason

Because - C & sup2; + A & sup2; + 2ab-2bc = 0
So B & sup2; + A & sup2; + 2Ab = B & sup2; + 2BC + C & sup2;
So (a + b) & sup2; = (B + C) & sup2;
And a + b > 0, B + C > 0
So a + B = B + C
So a = C
So it's an isosceles triangle
Let the first of four consecutive numbers be n, then
n(n+1)(n+2)(n+3)+1
=n(n+3)(n+1)(n+2)+1
=(n² +3n)(n² +3n+2)+1
=(n² +3n)²+2(n²+3n)+1
=(n²+3n+1)²
Therefore, the sum of the product of any four consecutive positive integers and one must be a complete square number

If the center of the ellipse is at the origin, the focus is on the x-axis, the line between a focus and the two ends of the minor axis is perpendicular to each other, and the distance between the focus and the nearer end of the major axis is 10 − 5, then the equation of the ellipse is as follows:______ ．

Let the equation of ellipse be x2a2 + y2b2 = 1 (a ＞ B ＞ 0). Since a focal point is perpendicular to the two ends of the minor axis, the distance from the focal point to the nearer end of the major axis of B = C is 10 − 5, so a-c = 10 − 5, ∵ A2 = B2 + C2 ℅ a = 10, B = C = 5. The equation of ellipse is: x210 + Y25 = 1, so the answer is: x210 + Y25 = 1

If we know that the parabola passes through the point (4, - 3), and x = 3, the function has the maximum value of 4, then its analytical formula is____

2
Let y = a (x-3) + 4 a

It is known that the parabola C: y = 2x ^ 2, the straight line y = KX + 2 intersects C at two points a and B, M is the midpoint of the line AB, and the vertical line passing through M as the X axis intersects C at n. It is proved that the straight line L passing through N and parabola C has only one intersection is parallel to ab

(1) Let a (x1, Y1), B (X2, Y2), x1 ＞ X2 (point a is on the right side of point B) substitute y = KX + 2 into y = 2x & # 178; - kx-2 = 0  X1 + x2 = K / 2, x1x2 = - 1

What's the formula for solving the problem of two cars meeting (all)

Basic concept: the problem of travel studies the motion of an object. It studies the relationship among speed, time and travel
Basic formula: distance = speed × time; distance △ time = speed; distance △ speed = time
Key problem: determine the position during the journey
Encounter problem: speed and X encounter time = encounter distance (please write other formulas)
Pursuit problem: pursuit time = distance difference △ speed difference (write other formulas)
Flow problem: downstream stroke = (ship speed + water speed) × downstream time, upstream stroke = (ship speed water speed) × upstream time
Forward speed = ship speed + water speed, backward speed = ship speed - water speed
Hydrostatic velocity = (downstream velocity + upstream velocity) × 2 water velocity = (downstream velocity - upstream velocity) × 2
Flow problem: the key is to determine the speed of the object, refer to the above formula
Bridge problem: the key is to determine the distance of the object, refer to the above formula
For reference only:
[formula of sum difference problem]
(sum + difference) △ 2 = larger number;
(sum difference) △ 2 = less decimal
[formula of sum multiple problem]
And (multiple + 1) = a multiple;
One multiple x multiple = another number,
Or sum - a multiple = another number
[formula of difference multiple problem]
Difference (multiple-1) = smaller decimal;
Smaller number × multiple = larger number,
Or smaller number + difference = larger number
[formula of average problem]
Total quantity △ total copies = average
[formula of general travel problem]
Average speed × time = distance;
Distance △ time = average speed;
Distance / average speed = time
[formula of reverse travel problem] the reverse travel problem can be divided into two types: "meeting problem" (two people start from two places and walk in opposite directions) and "separation problem" (two people walk in opposite directions). Both of these problems can be solved by the following formula
(speed and) × encounter (departure) time = encounter (departure) distance;
Distance of encounter (departure) / (speed sum) = encounter (departure) time;
Distance of meeting (leaving) and time of meeting (leaving) = speed and distance
[formula of travel in the same direction]
Catch up (pull out) distance (speed difference) = catch up (pull out) time;
Catch up (pull away) distance △ catch up (pull away) time = speed difference;
(speed difference) × overtaking time = overtaking distance
[formula of train crossing bridge problem]
(bridge leader + train leader) △ speed = bridge crossing time;
(bridge leader + train leader) △ bridge crossing time = speed;
Speed × crossing time = the sum of bridge and vehicle length
[formula of sailing problem]
(1) General formula:
Static water speed (ship speed) + current speed (water speed) = downstream speed;
Ship speed water speed = upstream speed;
(downstream speed + upstream speed) △ 2 = ship speed;
(downstream velocity - upstream velocity) 2 = water velocity
(2) The formula of two ships sailing in opposite directions:
Ship a's downstream speed + ship B's upstream speed = ship a's still water speed + ship B's still water speed
(3) The formula of two ships sailing in the same direction:
The static water velocity of fore (AFT) ship - the static water velocity of fore (AFT) ship = the speed of narrowing (widening) the distance between two ships
After calculating the speed of the distance between the two ships, answer the question according to the above formula
[engineering problem formula]
(1) General formula:
Work efficiency × working hours = total amount of work;
Total amount of work △ working hours = work efficiency;
Total amount of work △ work efficiency = working hours
(2) The formula for solving engineering problems by assuming that the total amount of work is "1" is as follows
1 △ working time = fraction of the total amount of work completed in unit time;
1 △ what percentage of work can be completed per unit time = working time
(Note: to solve engineering problems with hypothesis method, you can arbitrarily assume that the total amount of work is 2, 3, 4, 5 Especially when the total amount of work is assumed to be the least common multiple of several working hours, the fractional engineering problem can be transformed into a relatively simple integer engineering problem, and the calculation will become relatively simple.)
[profit and loss formula]
(1) If there is surplus (surplus) at one time and insufficient (deficit) at one time, the following formula can be used:
(profit + loss) / (the difference between the two distributions) = the number of people
For example, "each child has 10 peaches less than 9, and each child has 8 peaches more than 7. Question: how many children and how many peaches are there?"
Solution (7 + 9) / (10-8) = 16 / 2
=8 (pieces) Number of people
10 × 8-9 = 80-9 = 71 Peach
Or 8 × 8 + 7 = 64 + 7 = 71
(2) There is surplus (surplus) in two times
(big profit - small profit) / (the difference between the number of people allocated twice) = the number of people
For example, "when soldiers carry bullets for marching training, each person carries 45 rounds, an additional 680 rounds; if each person carries 50 rounds, an additional 200 rounds will be provided. Question: how many soldiers are there? How many bullets are there?"
Solution (680-200) △ 50-45) = 480 △ 5
=96 (person)
45 × 96 + 680 = 5000
Or 50 × 96 + 200 = 5000 (hair)
(3) Two times are not enough
(big loss - small loss) / (the difference between the number of people allocated twice) = the number of people
For example, "if you distribute a batch of books to students, each of them will receive 10 copies, with a difference of 90 copies; if each of them receives 8 copies, there will still be a difference of 8 copies. How many students and how many copies are there?"
Solution (90-8) / (10-8) = 82 / 2
=41 (person)
10 × 41-90 = 320
(4) One time is not enough (loss), and the other time is just finished
Deficit (the difference between the number of people allocated twice) = number of people
(example omitted)
(5) There is surplus (surplus) in one time, and it is just finished in the other time
Surplus (the difference between the two distributions) = number of people
(example omitted)
[formula of chicken rabbit problem]
(1) Given the total number of heads and feet, find the number of chickens and rabbits
(total feet - feet per chicken × total head) / (feet per rabbit - feet per chicken) = rabbits;
Total head count - Rabbit count = chicken count
Or (the number of feet per rabbit × the total number of heads - the total number of feet) / (the number of feet per rabbit - the number of feet per chicken) = the number of chickens;
Total head number chicken number = rabbit number
For example, "there are 36 chickens and rabbits. They have 100 feet. How many chickens and rabbits are there?"
Solution 1 (100-2 × 36) / (4-2) = 14 (pieces) Rabbits;
36-14 = 22 Chicken
Solution 2 (4 × 36-100) / (4-2) = 22 (pieces) Chicken;
36-22 = 14 Rabbit
(brief answer)
(2) When the total number of feet of chicken is more than that of rabbit, the formula can be used
(the number of feet per chicken × the difference between the total number of heads and the number of feet) / (the number of feet per chicken + the number of feet per rabbit) = the number of rabbits;
Total head rabbit = chicken
Or (the difference between the number of feet of each rabbit × the total number of heads + the number of feet of each rabbit) / (the number of feet of each chicken + the number of feet of each rabbit) = the number of chickens;
Total head number chicken number = rabbit number
(3) Given the difference between the total number and the number of feet of chickens and rabbits, the formula can be used when the total number of feet of rabbits is more than that of chickens
(the number of feet per chicken × the total number of heads + the difference between the number of feet per chicken and rabbit) / (the number of feet per chicken + the number of feet per rabbit) = the number of rabbits;
Total head count - Rabbit count = chicken count
Or (the number of feet of each rabbit × the difference between the total number of heads and the number of feet of chickens and rabbits) / (the number of feet of each chicken + the number of feet of each rabbit) = the number of chickens;
Total head number chicken number = rabbit number
(4) The following formula can be used to solve the problem of gain and loss
(score of 1 qualified product × total number of products actual total score) / (score of each qualified product + deduction score of each unqualified product) = number of unqualified products. Or total number of products - (deduction score of each unqualified product × total number of products + actual total score) / (score of each qualified product + deduction score of each unqualified product)