Let a > 0, b > 0, prove that (x + a) 2 (X-B) + x2 = 0 has one positive root and two negative roots (zero point theorem and intermediate value theorem of higher numbers)

Note: F (x) = [(x + a) ^ 2] (X-B) + x ^ 2
We know that f (x) is a cubic function, which is continuous on the whole number axis and has at most three zeros
f(0)= -(a^2)b 0,
f(b) = b^2 >0
X tends to - ∞ f (x) tends to - ∞,
In order:
-∞ -a,0 ,b ,
From the intermediate value theorem, we know that f (x) has at least one negative zero point in (- ∞), - A, (- A, 0) respectively,
In (0, b), there is at least one positive zero point
Since f (x) has at most three real zeros, it is deduced that:
F (x) has unique negative zeros in (- ∞), - A, (- A, 0) respectively,
In (0, b), there is only one positive zero point
It is proved that the equation: [(x + a) ^ 2] (X-B) + x ^ 2 = 0 has one positive root and two negative roots

Zero point theorem of higher numbers
Let f (x) d have f (x) - f (y) ≤ l X-Y for any two points x and Y on the closed interval [a, b], where l is a normal number and f (a) · f (b) < 0. It is proved that there is at least one point ξ Ε (a, b) such that f (ξ) = 0

Because f (a) · f (b) < 0, we only need to prove whether f (x) is continuous or not by using the zero point theorem, because ︱ f (x) - f (y) ≤ L ︱ X-Y ︱ suppose y = x + △ x, the original formula = ︱ f (x) - f (x + △ x) ≤ L ︱ X - (x + △ x) = l ︱ x ︱ so when ︱ x tends to 0, 0 ︱ f (x) - f (x + △ x) ≤ L ︱

What is the result of 1 + 2 + 3 + 4 +. + 998 + 999?

（1+999）×1000÷2
=5000000

The total weight of the three steel plates is 630kg. The weight of the first steel plate is three times that of the second steel plate, and the weight of the second steel plate is two times that of the third steel plate?

The third block is 1, the second block is 2, and the first block is 2 * 3 = 6
Third piece: 630 / (1 + 2 + 6) = 70kg
Second piece: 70 * 2 = 140kg
First piece: 14083 = 420 kg

If AB = - 89, find the value of 3A + 2BB

∵ab=-89，∴a=-89b，∴3a+2bb=3×(−89b)+2bb=-83+2=-23．

Let X1 and X2 be the two complex roots of the quadratic equation x ^ 2 + X + P = 0 with real coefficients, and find the two roots of the equation and the corresponding value of P

X ^ 2 + X + P = 0 has imaginary root and △ = 1-4p

In the primary school fruit shop, apples, bananas and peaches weigh seven eighths of a ton, of which apples and bananas weigh three fourths
Apples, bananas and peaches in the fruit shop weigh seven eighths of a ton, of which apples and bananas weigh three fourths of a ton, bananas and peaches weigh one-half of a ton. How many tons of bananas are there (can you answer them in different ways?)
I also know that the answer is: "half + three fourths - seven eighths = three eighths of a ton". But please tell me what I think and how to calculate

This problem is very simple. We already know that "apples, bananas and peaches weigh seven eighths of a ton", and "apples and bananas weigh three fourths of a ton", then peaches are seven eighths minus three fourths, which is equal to one eighth. Similarly, "bananas and peaches weigh one half of a ton", then apples are seven eighths minus one half, which is equal to three eighths, So banana is the sum of the three, 7 / 8 minus 1 / 8 minus 3 / 8 equals 3 / 8
Another way is to let x, y, Z form, but the principle is the same as above
Your method is good. It's good. My method is more complicated,
You can also subtract three fourths from seven eighths to get the weight of the peach, and then subtract the weight of the peach from the banana and one half of the peach to get the weight of the banana

In a right triangle, angle c is equal to 90 degrees, B = 3, angle a is equal to 30 degrees, find a, C

If the direction vector a of the line L = (- 2,3,1) and a normal vector n of the plane z = (4,0,1), then the cosine of the angle between the line L and the plane Z is the whole process

Let the angle between the line L and the plane Z be θ
∴sinθ=│（a·n/│a│·│n│）│
=│-7/√14×√17│
=√34/34
∵θ∈[0,π/2]
∴cosθ=√1022/34

Hongxing primary school organized students to walk in line for an outing, 60 meters per minute. Teacher Wang at the end of the line arrived at the front of the line at the speed of 150 meters per minute,
Then it took 10 minutes to return immediately. How long was the team?
It's better to be before 9:00 this evening,
It's urgent to count!

Mr. Wang back and forth to walk the relative distance is the length of the team
Relative speed to the front row 150-60 = 90
At the end of the row, the relative velocity is 150 + 60 = 210
So the time ratio is s / 90: S / 210 = 7:3
It takes 7 minutes to get to the front row and 3 minutes to return
Team length s = 90 * 7 = 210 * 3 = 630 M

Let a > 0, b > 0, prove that (x + a) 2 (X-B) + x2 = 0 has one positive root and two negative roots (zero point theorem and intermediate value theorem of higher numbers)