a) 2x²- 6x²

= -4x²

b) x²-6x+9-y²

= (x-3)² -y²

= (x-3-y).(x-3+y)

đổi 300kg = 3000N

ta có công thức: F x OA = P x OB

                       3000 x 40 = 800 x OB

\(\Rightarrow OB=\dfrac{3000\cdot40}{800}=150\left(cm\right)\)

chiều dài đòn bẩy tối thiểu là:

AB = OA + OB = 40 + 150 = 190 (cm)

Lời giải:
Ta có:

$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)[(a^2+ab+b^2)-2ab]$

Áp dụng BĐT AM-GM:

$a^2+ab+b^2=(a^2+b^2)+ab\geq 2ab+ab=3ab$

$\Rightarrow 2ab\leq \frac{2(a^2+ab+b^2)}{3}$

$\Rightarrow a^2-ab+b^2=a^2+b^2+ab-2ab\geq a^2+b^2+ab- \frac{2}{3}(a^2+ab+b^2)=\frac{1}{3}(a^2+ab+b^2)$

$\Rightarrow a^3+b^3=(a+b)(a^2-ab+b^2)\geq \frac{1}{3}(a+b)(a^2+ab+b^2)$

$\Rightarrow \frac{a^3+b^3}{a^2+ab+b^2}\geq \frac{1}{3}(a+b)$

Hoàn toàn tương tự với các phân thức khác và cộng theo vế thu được:

$P\geq \frac{1}{3}(a+b)+\frac{1}{3}(b+c)+\frac{1}{3}(c+a)=\frac{2}{3}(a+b+c)$

$\geq \frac{2}{3}.3\sqrt[3]{abc}=2$

Vậy $P_{\min}=2$. Giá trị này đạt tại $a=b=c=1$

Lời giải:
Ta có:

$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)[(a^2+ab+b^2)-2ab]$

Áp dụng BĐT AM-GM:

$a^2+ab+b^2=(a^2+b^2)+ab\geq 2ab+ab=3ab$

$\Rightarrow 2ab\leq \frac{2(a^2+ab+b^2)}{3}$

$\Rightarrow a^2-ab+b^2=a^2+b^2+ab-2ab\geq a^2+b^2+ab- \frac{2}{3}(a^2+ab+b^2)=\frac{1}{3}(a^2+ab+b^2)$

$\Rightarrow a^3+b^3=(a+b)(a^2-ab+b^2)\geq \frac{1}{3}(a+b)(a^2+ab+b^2)$

$\Rightarrow \frac{a^3+b^3}{a^2+ab+b^2}\geq \frac{1}{3}(a+b)$

Hoàn toàn tương tự với các phân thức khác và cộng theo vế thu được:

$P\geq \frac{1}{3}(a+b)+\frac{1}{3}(b+c)+\frac{1}{3}(c+a)=\frac{2}{3}(a+b+c)$

$\geq \frac{2}{3}.3\sqrt[3]{abc}=2$

Vậy $P_{\min}=2$. Giá trị này đạt tại $a=b=c=1$

Bài 2

a) A = (-2x²y² + 4xy - 6xy³) : 2xy

= -xy + 2 - 3y²

Thay x = 1/2; y = 4 vào A, ta có:

A = -1/2 . 4 + 2 - 3 . 2²

= -2 + 2 - 12

= -12

b) B = 25x² - 10xy² + y⁴

= (5x - y²)²

Thay x = 2; y = 3 vào B, ta có:

B = (5.2 - 3²)²

= (10 - 9)²

= 1²

= 1

c) C = (3x + 2)² + 2(3x + 2)(2y - 1)² + (2y - 1)²

= (3x + 2 + 2y - 1)²

= (3x + 2y + 1)²

Thay x = 1/3; y = -1/2 vào C, ta có:

C = [3 . 1/3 + 2 . (-1/2) + 1]²

= (1 - 1 + 1)²

= 1²

= 1

Bài 3

a) x³ + 2x² = x²(x + 2)

b) 3(x - y) - 5x(y - x)

= 3(x - y) + 5x(x - y)

= (x - y)(3 + 5x)

c) 4x³ - 9x

= x(4x² - 9)

= x(2x - 3)(2x + 3)

d) (x - 2y)² - 4(x + y)²

= (x - 2y)² - [2(x + y)]²

= (x - 2y) - (2x + 2y)²

= (x - 2y - 2x - 2y)(x - 2y + 2x + 2y)

= (-x - 4y).3x

= -3x(x + 4y)

e) x²y + x² - 4y - 4

= (x²y + x²) - (4y + 4)

= x²(y + 1) - 4(y + 1)

= (y + 1)(x² - 4)

= (y + 1)(x - 2)(x + 2)

f) -27x³(x + 1) + x + 1

= - 27x³(x + 1) + (x + 1)

= (x + 1)(-27x³ + 1)

= (x + 1)(1 - 27x³)

= (x + 1)(1 - 3x)(1 + 3x + 9x²)

b, 9a² - 6a + 1 - 25b²

= (3a - 1)² - (5b)²

= (3a - 1 - 5b).(3a -1 + 5b)

b,

B =  \(\dfrac{1}{x+2}\) + \(\dfrac{5}{x-2}\) - \(\dfrac{2x}{x^2-4}\) (đk \(x\) ≠ -2; 2)

B = \(\dfrac{1}{x+2}+\dfrac{5}{x-2}-\dfrac{2x}{\left(x-2\right)\left(x+2\right)}\)

B = \(\dfrac{x-2+5.\left(x+2\right)}{\left(x+2\right)\left(x-2\right)}\) - \(\dfrac{2x}{\left(x-2\right)\left(x+2\right)}\)

B = \(\dfrac{x-2+5x+10-2x}{\left(x+2\right).\left(x-2\right)}\)

B = \(\dfrac{4x+8}{\left(x-2\right)\left(x+2\right)}\)

B = \(\dfrac{4}{x-2}\)

C,  

C = \(\dfrac{1}{x+1}\) + \(\dfrac{2}{1-x}\) - \(\dfrac{1-5x}{x^2-1}\) Đk \(x\ne\) -1; 1

C = \(\dfrac{x-1}{\left(x+1\right)\left(x-1\right)}\) - \(\dfrac{2.\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\) - \(\dfrac{1-5x}{\left(x-1\right)\left(x+1\right)}\)

C = \(\dfrac{x-1-2x-2-1+5x}{\left(x-1\right)\left(x+1\right)}\)

C = \(\dfrac{-4x-4}{\left(x+1\right)\left(x-1\right)}\)

C = \(\dfrac{-4\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)

C = \(\dfrac{-4}{x-1}\)

Chọn B

Remarkable: đáng chú ý 

Have you ever been to Britain? I’ve dreamed of going there (1) __ I was a little girl and finally, (2) ___ the spring of 2012, it happened. I was there three weeks and (3) ___much of that time soaking up the culture, history, and scenery of this incredible island. I visited several districts in England, Wales and Scotland and even (4) ___ a site trip to Ireland. Every place I went to seemed to be more interesting and more beautiful (5) ____the last. The tour I was with began in London. Of all the sights I saw there, the Tower of London was the most (6) ___The Tower is a building complex of incredibly rich history.

Also on the Thames are the Houses of Parliament and, of course, the clock tower housing “Big Ben”. I saw so many landmarks I can hardly remember them all. Although it was just a glimpse of London I did, (7) __, get a chance to see all the (8)____ famous places.

After (9) ___ of days in London we traveled to the south of England stopping to visit Stonehenge. If you don’t know about Stonehenge there are a couple of links on my links page that will (10) ___ you to sites that can fill you in.

   1. A. for                        B. while                     C. since                      D. because

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   8. A. more                    B. much                     C. most                       D. many

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