a,

ta có:

(x²+7x+3)²=x⁴+14x³+55x²+42x+9

(8x+4)(x²+5x+2)=8x³+44x²+36x+8

=>x⁴+14x³+55x²+42x+9=8x³+44x²+36x+8

<=>x⁴+6x³+11x²+6x+1=0

xét x=0 ko phải no của pt

xét x khác 0

\(\Leftrightarrow\left(x^2+\frac{1}{x^2}\right)+6\left(x+\frac{1}{x}\right)+11=0\)

\(\Leftrightarrow\left(x+\frac{1}{x}\right)^2+6\left(x+\frac{1}{x}\right)+9=0\Leftrightarrow\left(x+\frac{1}{x}+3\right)^2=0\Rightarrow x=\frac{-3+\sqrt{5}}{2};\frac{-3-\sqrt{5}}{2}\)

d,

xét n=1=> mệnh đề luôn đúng

giả sử mệnh đề đúng với n=k

ta sẽ cm nó đúng với n=k+1

với n=k+1

=>(n+1)(n+2)..(n+n)=2n(n+1)(n+2)...(2n-1)

=2(k+1)(k+2).....2k chia hết cho 2^k+1

=>(n+1)(n+2)(n+3)...(n+n) chia hết cho 2ⁿ

c,

ta có:

\(\left(1+x\right)\left(1+\frac{y}{x}\right)=1+x+y+\frac{y}{x}\ge1+y+2\sqrt{y}=\left(\sqrt{y}+1\right)^2\)

\(\Rightarrow\left(1+x\right)\left(1+\frac{y}{x}\right)\left(1+\frac{9}{\sqrt{y}}\right)^2\ge\left[\left(\sqrt{y}+1\right)\left(1+\frac{9}{\sqrt{y}}\right)\right]^2\)

\(=\left(\sqrt{y}+\frac{9}{\sqrt{y}}+10\right)^2\ge\left(6+10\right)^2=256\left(Q.E.D\right)\)

dấu = xảy ra khi y=9;x=3

b,

x⁷+xy⁶=y¹⁴+y⁸

<=>(x⁷-y¹⁴)+(xy⁶-y⁸)=0

<=>(x-y²)(x+y²)+y⁶(x-y²)=0

<=>(x-y²)(x+y²+y⁶)=0

xét x=y²

\(\Rightarrow\sqrt{4x+5}+\sqrt{y^2+8}=\sqrt{4y^2+5}+\sqrt{y^2-1}\)

\(\Rightarrow\sqrt{4y^2+5}+\sqrt{y^2+8}=6\)

\(\Rightarrow\left(\sqrt{4y^2+5}-3\right)+\left(\sqrt{y^2+8}-3\right)=0\)

\(\Rightarrow\frac{4y^2-4}{\sqrt{4y^2+5}+3}+\frac{y^2-1}{\sqrt{y^2+8}+3}=0\)

\(\Rightarrow\left(y^2-1\right)\left(\frac{4}{\sqrt{4y^2+5}+3}+\frac{1}{\sqrt{y^2+8}+3}\right)=0\)

\(\frac{4}{\sqrt{4y^2+5}+3}+\frac{1}{\sqrt{y^2+8}+3}>0\Rightarrow y^2=1\Rightarrow\left(x;y\right)=\left(1;1\right);\left(1;-1\right)\)

xét x+y²+y⁶=0

<=>x=-y²-y⁶

lại có:

x⁷+xy⁶=y¹⁴+y⁸

<=>x(x⁶+y⁶)=y¹⁴+y⁸

<=>-(y²+y⁶)(x⁶+y⁶)=y¹⁴+y⁸

mà \(-\left(y^2+y^6\right)\left(x^6+y^6\right)\le0\le y^{14}+y^8\)

<=>y=0=>x=0(ko thỏa mãn)

vậy nghiệm của pt:(x;y)=(1;-1);(1;1)

Hoàn thành chuyển đổi sau:

         (1)    (2)    (3)             (4)    (5)    (6)     (7)

\(FeS_2\rightarrow A\rightarrow B\rightarrow H_2SO_4\rightarrow A\rightarrow D\rightarrow C\rightarrow A\)

        (8)   \(\downarrow\)            (9)   \(\downarrow\) 

               \(C\)                   \(E\rightarrow BaSO_4\)

                                        (10)

Tìm các giới hạn sau:

\(\lim\limits_{x\rightarrow-\infty}\) \(\dfrac{\sqrt{x^6+2}}{3\text{x}^3-1}\)

\(\lim\limits_{x\rightarrow+\infty}\) \(\dfrac{\sqrt{x^6+2}}{3\text{x}^3-1}\)

\(\lim\limits_{x\rightarrow-\infty}\) \(\left(\sqrt{2\text{x}^2+1}+x\right)\)

\(\lim\limits_{x\rightarrow1}\) \(\dfrac{2\text{x}^3-5\text{x}-4}{\left(x+1\right)^2}\)

Tìm giá trị nhỏ nhất : \(\frac{x^2+6x-22}{5}\)

\(\frac{x^2+6x-22}{5}=\frac{x^2+6x+9-9-22}{5}=\frac{\left(x+3\right)^2-31}{5}\)

Đa thức nhỏ nhất \(\Leftrightarrow\left(x+3\right)^2-31\)nhỏ nhất

\(\Leftrightarrow\left(x+3\right)^2=0\Rightarrow x+3=0\Rightarrow x=-3\)

\(\Rightarrow\)đa thức nhỏ nhất \(=\frac{-31}{5}\)\(\Leftrightarrow x=-3\)

Ké chút ạ

ex1:Write sentences with the cue words.Use the present perfect tense of verbs.

0.I /know/her/three months.\(\Rightarrow\)I have known her for three months.

1.My sister/have/a red bike/yesterday.\(\rightarrow\)...

2.I/not eat/in that restaurant/ages.\(\rightarrow...\)

3.Mr.Robinson/teach/in this school/1987.\(\rightarrow...\)

4.I/write/this letter/six o'clock.\(\rightarrow...\)

ex2.Put the verbs in brackets in the correct verb tense.

1.Be careful.The teacher (look) . . . at you.

2.Our school performance (start) . . .late last Sunday because of the heavy rain.

3.The boy (learn) . . .for three years,but he can't understand this letter.

4.. . .(she/feed). . .the cat yet ?

5.. . .(your dog/ever bite). . .anyone ?

6.. . .(you/ever be). . .to Ha Noi ?

Finish each of the following sentences in such a way that it means exactly the same as the sentence printed before it.

1. We started cooking for the party four hours ago.

\(\rightarrow\) We ...........................................................................

2. Learning English is quite easy.

\(\rightarrow\) It is ............................................................................

3.Many people speak English all over the world.

\(\rightarrow\) English........................................................................

4. The yard is too small to play soccer in.

\(\rightarrow\) The yard isn't ................................................................

5. She is very pleased to see her parents soon.

\(\rightarrow\) She is looking ...........................................................

6. "Help me to hang the picture" Mai said to Ba.

\(\rightarrow\) Mai asked ..............................................................

7. "I will come to see you as soon as I can" Lan said.

\(\rightarrow\) Lan said ...................................................................

8. It was such good weather that we could go swimming.

\(\rightarrow\) The weather .............................................................

9. I would like you to help me to put the chair away .

\(\rightarrow\) Would you mind ............................................................................

10. Who wrote this book.

\(\rightarrow\) Who ...........................................................................