\(\sqrt{\dfrac{a+b}{2}}\ge\dfrac{\sqrt{a}+\sqrt{b}}{2}\)

\(\Leftrightarrow\dfrac{a+b}{2}\ge\dfrac{a+2\sqrt{ab}+b}{4}\)

\(\Leftrightarrow4a+4b\ge2a+4\sqrt{ab}+2b\)

\(\Leftrightarrow2a+2b\ge a+2\sqrt{ab}+b\)

\(\Leftrightarrow a-2\sqrt{ab}+b\ge0\)

\(\Leftrightarrow\left(\sqrt{a}-\sqrt{b}\right)^2\ge0\) (luôn đúng)

Đẳng thức xảy ra \(\Leftrightarrow a=b\)

Ta có: điều phải chứng minh

1. B

2. A

3. C

4. D

5. C

6. D

7. C

8. A

9. B

10. B

11. B

12. D

13. A

14. D

15. D

c) \(\dfrac{6+3^2.2-6.3^2}{3^2+3.3^2-3^4}=\dfrac{3.2.1+3.2.3-3.2.3^2}{3^2.1+3^2.3-3^2.3^2}=\dfrac{3.2\left(1+3-3^2\right)}{3^2\left(1+3-3^2\right)}=\dfrac{2}{3}\)

d) \(\dfrac{5^{10}.7^3-25^5.49^2}{\left(125.7\right)^3+5^9.14^3}=\dfrac{5^{10}.7^3-\left(5^2\right)^5.\left(7^2\right)^2}{125^3.7^3+\left(5^3\right)^3.\left(7.2\right)^3}=\dfrac{5^{10}.7^3-5^{10}.7^4}{\left(5^3\right)^3.7^3+\left(5^3\right)^3.7^3.2^3}=\dfrac{5^{10}.7^3-5^{10}.7^3.7}{\left(5^3\right)^3.7^3+\left(5^3\right)^3.7^3.8}=\dfrac{5^{10}.7^3\left(1-7\right)}{\left(5^3\right)^3.7^3\left(1+8\right)}=\dfrac{5^{10}.7^3.\left(-6\right)}{5^9.7^3.9}=\dfrac{-10}{3}\)

2x² - x - 6 = 0

<=> (2x² - 4x) + (3x - 6) = 0

<=> 2x(x - 2) + 3(x - 2) = 0

<=> (2x + 3)(x - 2) = 0

<=> 2x + 3 = 0 hoặc x - 2 = 0

\(\Leftrightarrow x=-\dfrac{3}{2}\) hoặc x = 2

Vậy phương trình có tập nghiệm là: \(S=\left\{-\dfrac{3}{2};2\right\}.\)

1. The building where he lives is very old.

2. I will tell you something which you didn't know it before.

3. I'll never forget the day when I met you.

4. Although it was cold, Tom still went swimming.

5. John is very sad because he fails the final exam.

6. He never left the house where he had been born.

7. I met the man who works in a bank.

8. Here are the letters which arrived this morning.

9. Although my grandfather is very old, he is still young at heart.

10. Tom gets a lot of money because he works hard.

Biến đổi vế trái, ta có:

\(\sqrt[3]{26+15\sqrt{3}}\)

\(=\sqrt[3]{8+12\sqrt{3}+18+3\sqrt{3}}\)

\(=\sqrt[3]{8+3.2.2\sqrt{3}+3.2.3+3\sqrt{3}}\)

\(=\sqrt[3]{2^3+3.2^2.\sqrt{3}+3.2.\left(\sqrt{3}\right)^2+\left(\sqrt{3}\right)^3}\)

\(=\sqrt[3]{\left(2+\sqrt{3}\right)^3}\)

\(=2+\sqrt{3}\) (điều phải chứng minh)

1. celebration

2. world

3. reunion

4. local

5. ancestors

6. biggest

7. made

8. traditional

9. mooncakes

10. money

If Daisy studies harder, she will pass the examination.

\(\sqrt{3x-2}=x-1\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-1\ge0\\3x-2=\left(x-1\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\3x-2=x^2-2x+1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\x^2-5x+3=0\end{matrix}\right.\)

Ta có: \(\Delta=5^2-4.1.3=25-12=13>0\Rightarrow\) Phương trình có hai nghiệm phân biệt là: \(\left[{}\begin{matrix}x=\dfrac{5+\sqrt{13}}{2}\\x=\dfrac{5-\sqrt{13}}{2}\end{matrix}\right.\)

So với điều kiện xác định thì \(x=\dfrac{5+\sqrt{13}}{2}\) thỏa mãn.

Vậy phương trình có nghiệm duy nhất là \(x=\dfrac{5+\sqrt{13}}{2}.\)

1. would come

2. being arrested

3. was destroyed

4. is often taken

5. were sleeping

6. stay

7. to speak

8. comes

9. hasn't been used

10. are cleaned