The difference between 8 times of a number and 2,4 is exactly 12. What is the number

The difference between 8 times of a number and 2,4 is exactly 12. What is the number

The difference between 8 times of a number and 2,4 is exactly 12. This number is (1.8)
（2.4+12）÷8
=14.4÷8
=1.8
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Known: as shown in the figure, m (3,2), n (1, - 1). Point P makes PM + PN shortest on the y-axis, and calculates the coordinates of point P

Draw a figure according to the meaning of the question, find out the symmetrical point n 'of point n about y axis, connect Mn', and the intersection point with y axis is the point P, ∵ n (1, - 1), ∵ n '(- 1, - 1). Let the analytical expression of straight line Mn' be y = KX + B, substitute m (3,2), n '(- 1, - 1) into the result: 3K + B = 2 − K + B = − 1, and the solution is k = 34B = − 14, so y = 34x-14, let x = 0, and y = - 14, then the point P coordinate is（ 0，−14）．

Given a ^ 2-3a-1 = 0, find the value of: ① a ^ 2 + 1 / (a ^ 2); ② a ^ 3-1 / (a ^ 3); ③ a ^ 3 + 1 / (a ^ 3)
A difficult math problem 3q in Junior Three

First, we get a ≠ 0, so the following formulas are meaningful
a²-3a+1=0
a²+1=3a
Square on both sides
a^4+2a²+1=9a²
a^4+1=7a²
Divide both sides by a & sup2;
a²+1/a²=7
①a²+1/a²=7
②a³-1/a³=(a - 1/a)(a²+1+1/a²)
From a & sup2; - 3a-1 = 0, we can get a-3-1 / a = 0, A-1 / a = 3
So ② = 3 × 8 = 24
③a³+1/a³=(a+1/a)(a²-1+1/a²)
a-1/a=3,(a-1/a)²=9,(a-1/a)²+4=13,a²+1/a²+2=13,a+1/a=±√13
So ③ = ± 6 √ 13
Third, I don't know if it's right. The first two are definitely right
But look at the steps should be no problem~
If you have any questions, you can go to Baidu Hi

As shown in the figure, in the right angle trapezoid ABCD, ad ‖ BC, ∠ B = 90 ° and AD + BC = CD (1) take CD as the diameter, make circle O, and prove that AB is tangent to circle o;
(2) Taking AB as the diameter of the circle O ', it is proved that CD is tangent to the circle O'

(1) through O, OE ⊥ AB is set at e, ∵ a = ∠ B = 90 °, and ∥ ad ∥ OE ∥ BC,
∵ o is the midpoint of CD, ∵ e is the midpoint of ab,
OE = 1 / 2 (AD + BC) = 1 / 2CD = radius,
⊙ AB is tangent to ⊙ o
(2) connect do 'to CB extension line F,
∫ ad ∥ BC, ∫ o'ad = ∫ o'bf, ∫ o'da = ∫ o'fb, OA = ob,
∴ΔO'AD≌ΔO'BF,∴O'D=O'B',AD=BF,
∴CD=AD+BC=BF,
∴CO'⊥DF,∠O'CD=∠O'CB,
If we make o'g ⊥ CD through o ', then o'g = o'b,
⊙ CD is the tangent of ⊙ o '
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Given the equations x ^ 2 + M1 x + N1 = 0 and x ^ 2 + M2 x + N2 = 0, and M1 M2 = 2 (N1 + N2), it is proved that at least one of the two equations has a real root

To the contrary
If both equations have no real roots, then
delta1=m1^2-4n1

It is known that the radius of circumscribed circle O of quadrilateral ABCD is 2, the intersection of diagonal lines AC and BD is e, AE = CE, ab = root 2ae, and BD = 2 times root 3

∵AE*AC=AE*2AE=2AE^2=AB^2
∴AE：AB=AB：AC
Also ∵ ∠ BAC = ∠ EAB (common angle)
∴△ABE∽△ABC
∴∠ABD=∠ACB=∠ADB
∴AB=AD
The radius of circle O is 2
∴∠BAD=120°
∴∠ABD=∠ACB=∠ADB=30°
The height on the edge of BD = 1
∵AE=CE
The height of Δ abd is equal to that of Δ BCD
The area of quadrilateral ABCD = 2 △ the area of abd = 2 × (2 times root sign 3) × 1 △ 2 = 2 root sign 3

If f (x) = x & # 178; - 4mx + 5 is an increasing function on (- 2, + ∞), then the value range of M is

a=1>0
So on the right side of the axis of symmetry, it's an increasing function
The axis of symmetry x = - B / 2A = 2m

On the mathematical problems of plane rectangular coordinate system
In a plane rectangular coordinate system, there are two points a (1,5) B (6,1). Connect AB and make the vertical bisector of ab. find the analytical formula of the vertical bisector

According to the two-point formula, the analytic formula of the line AB can be obtained
(y-5)/(1-5)=(x-1)/(6-1)
The result is: y = (- 4 / 5) x + (29 / 5)
The abscissa of the midpoint o of line AB is: 1 + (6-1) / 2 = 7 / 2
Substituting x = 7 / 2 into y = (- 4 / 5) x + (29 / 5) yields y = 3
Therefore, the coordinates of the midpoint o of AB are o (7 / 2,3)
Because the product of the slopes of two vertical lines is - 1
So the slope of the straight line is 5 / 4, so the analytical formula of the straight line is y = (5 / 4) x + B
Because o (7 / 2,3) is also on the line, so
3=（5/4）*(7/2)+b
The solution is: B = - 11 / 8
Therefore, the analytical formula of the vertical bisector is: y = (5 / 4) x - (11 / 8)

Are there constants P and Q such that X4 + PX2 + Q can be divisible by x2 + 2x + 5? If it exists, find out the value of P and Q, otherwise, explain the reason

If there exists, then it is shown that X4 + PX2 + Q can be divided by x 2 + 2x + 5, and another factor is x2 + MX + N, and we can set another factor is x 2 + MX + N, {(x2 + 2x + 5) (x2 + 2x + 5) (x2 + 2x + 2x + 5) (x2 + MX2 + Q, that is, there is X4 + (M + 2) X3 + (M + 2 + 2m + 5) x 2 + (n + 2m + 2 + 2m + 5 + 2 + 2m + 5m) x + 5 N = n + 2m + 2m + 2 = 0n + 2m + 2m + 2m + 5 = P and 2n + 2m + 5m = 2n + 5 = 2n + 5 = 2n + 5m = 2n + 5 N = 05n = q, and 2n + 5m = 2n + 5m = 2n + 5m = 05n = 2n + 5m = 05n = 05n = 05n = 05n = q solve the above equations, and the above equations, we solve the it's not easy Find P = 6, q = 25

In the cube abcd-a1b1c1d1, where e is the midpoint of dd1, the positional relationship between BD1 and the plane passing ace is ()
A. Intersection B. parallel C. vertical D. line in plane

Connect AC and BD, the intersection point is f, and connect EF ∵ in △ bdd1, e and F are the midpoint of dd1 and BD, so EF ∥ BD1, ∥ EF ⊂ plane ace, BD1 ⊄ plane ace, ∥ BD1 ⊂ plane ace, so select B