làm tương tự bài này nha

x + y + z = 3. Tìm Max P = xy + yz + xz

Ta có: (x - y)² ≥ 0 <=> x² - 2xy + y² ≥ 0 <=> x² + y² ≥ 2xy

hay 2xy ≤ x² + y² , dấu " = " xảy ra <=> x = y 

tương tự: 

+) 2yz ≤ y² + z² +) 2xz ≤ x² + z² 

cộng 3 vế của 3 bđt trên

--> 2xy + 2yz + 2xz ≤ 2(x² + y² + z²) 

--> xy + yz + xz ≤ x² + y² + z² 

--> xy + yz + xz + 2xy + 2yz + 2xz ≤ x² + y² + z² + 2xy + 2yz + 2xz 

--> 3(xy + yz + xz) ≤ (x + y + z)² 

--> 3(xy + yz + xz) ≤ 3² 

--> xy + yz + xz ≤ 3 

Theo đề ta có :

xy + yz + xz = 0 

\(\Rightarrow xy=0-yz-xz=-\left(yz+xz\right)\) (1)

\(\Rightarrow yz=0-xz-xy=-\left(xz+xy\right)\)(2)

\(\Rightarrow xz=0-xy-yz=-\left(xy+yz\right)\)(3)

\(M=\frac{yz}{x^2}+\frac{xz}{y^2}+\frac{xy}{z^2}\)

Từ (1) ; (2) và (3) , ta có :

\(M=\frac{-\left(xy+xz\right)}{x^2}+\frac{-\left(xy+yz\right)}{y^2}+\frac{-\left(yz+xz\right)}{z^2}\)

\(M=\frac{-x\left(y+z\right)}{x^2}+\frac{-y\left(x+z\right)}{y^2}+\frac{-z\left(x+y\right)}{z^2}\)

\(M=\frac{-\left(y+z\right)}{x}+\frac{-\left(x+z\right)}{y}+\frac{-\left(x+y\right)}{z}\)

\(M-3=\left(\frac{-\left(y+z\right)}{x}-1\right)+\left(\frac{-\left(x+z\right)}{y}-1\right)+\left(\frac{-\left(x+y\right)}{z}-1\right)\)

\(M-3=\left(\frac{-y-z}{x}-\frac{x}{x}\right)+\left(\frac{-x-z}{y}-\frac{y}{y}\right)+\left(\frac{-x-y}{z}-\frac{z}{z}\right)\)

\(M-3=\left(\frac{-y-z-x}{x}\right)+\left(\frac{-x-z-y}{y}\right)+\left(\frac{-x-y-z}{z}\right)\)

\(M-3=\frac{-\left(y+z+x\right)}{x}+\frac{-\left(x+z+y\right)}{y}+\frac{-\left(x+y+z\right)}{z}\)

..............

Cái này quá dễ : 

\(\sqrt{\left(x-2\right)^2}=4\)

\(\Leftrightarrow\left(x-2\right)=4\)

=> x = 6

b) \(\sqrt{x^2+6x+9}=6\)

\(\Rightarrow\sqrt{\left(x+3\right)^2}=6\)

\(\Rightarrow x+3=6\)

=> x = 3 

a) \(\sqrt{\left(x-2\right)^{^2}}=4\)

<=> (x-2)² = 16 

TH1: x-2 = 4 <=> x= 6

TH2: x-2=-4 <=> x= -2

b) \(\sqrt{x^2+6x+9}=6\)

<=> x² + 6x+9=36

<=> (x+3)²=36

TH1: x+3 = 6 <=> x= 3

TH2: x+3 = -6 <=> x=-9

Thế này mới đủ nha :))

Ta có : \(\sqrt{3}.x-\sqrt{75}=0\)

\(\Leftrightarrow\sqrt{3}.x-5\sqrt{3}=0\)

\(\Leftrightarrow\sqrt{3}\left(x-5\right)=0\)

Vì \(\sqrt{3}\ne0\)

Nên : x - 5 = 0

Vậy x = 5. 

b) Ta có : \(\sqrt{2}.x+\sqrt{2}=\sqrt{8}+\sqrt{32}\)

\(\Leftrightarrow\sqrt{2}\left(x+1\right)=6\sqrt{2}\)

\(\Leftrightarrow\sqrt{2}\left(x+1\right)-6\sqrt{2}=0\)

\(\Leftrightarrow\sqrt{2}.\left(x+1-6\right)=0\)

\(\Leftrightarrow\sqrt{2}.\left(x-5\right)=0\)

Vì \(\sqrt{2}\ne0\)

Nên x - 5 = 0

Suy ra : x = 5

Ta có : \(A=\sqrt{\left(x+3\right)^2}-\frac{\sqrt{x^2-6x+9}}{x+3}\)

\(\Rightarrow A=\left(x+3\right)-\frac{\left(x-3\right)^2}{\left(x+3\right)}\)

\(\Rightarrow A=\frac{\left(x+3\right)^2}{\left(x+3\right)}-\frac{\left(x-3\right)^2}{\left(x+3\right)}\)

\(\Rightarrow A=\frac{\left(x+3\right)^2-\left(x-3\right)^2}{\left(x+3\right)}\)

\(\Rightarrow A=\frac{\left(x+3+x-3\right)\left(x+3-x+3\right)}{\left(x+3\right)}\)

\(\Rightarrow A=\frac{18x}{\left(x+3\right)}\)

Mình quên ghi điều kiện mất là x<-3 nhé

Mấy bạn làm lại giúp mình

Mình cảm ơn  :D

a) \(\sqrt{1\frac{9}{16}\times2\frac{14}{25}}=\sqrt{\frac{25}{16}\times\frac{64}{25}}=\sqrt{4}=2\)

b) \(\sqrt{\frac{25^2-9^2}{68}}=\sqrt{\frac{\left(25-9\right)\left(25+9\right)}{68}}=\sqrt{\frac{16.34}{68}}=\sqrt{8}\)

a) \(\sqrt{36-25}=\sqrt{11}\)

   \(\sqrt{36}-\sqrt{25}=6-5=1\)

 Suy ra \(\sqrt{36-25}>\sqrt{36}-\sqrt{25}\)

a,\(\sqrt{36-25}=-1\)

\(\sqrt{36}-\sqrt{25}=1\)

Vậy: \(\sqrt{36-25}< \sqrt{36}-\sqrt{25}\)

a. Ta có:\(\frac{x}{y}\sqrt{\frac{y^2}{x^4}=}\) \(\frac{x}{y}.\frac{\left|y\right|}{x^2}=\frac{x.y}{x^2y}\)\(=\frac{1}{x}\)(Vì \(x\ne0;y>0\))

b \(3x^2\sqrt{\frac{8}{x^2}}=3x^2\frac{2\sqrt{2}}{\left|x\right|}=\frac{6x^2\sqrt{2}}{-x}=-6x\sqrt{2}\)( Vì \(x< 0\))

\(\sqrt[3]{1+\sqrt{65}}-\sqrt[3]{\sqrt{65}-1}=\sqrt[3]{1+\sqrt{65}}+\sqrt[3]{1-\sqrt{65}}\).

Đặt \(a=\sqrt[3]{1+\sqrt{65}}\); \(b=\sqrt[3]{1-\sqrt{65}}\). Ta có: \(\hept{\begin{cases}a^3+b^3=2\\ab=-4\end{cases}}\)Suy ra:

\(\left(a+b\right)^3=2-12\left(a+b\right)\Leftrightarrow\left(a+b\right)^3+12\left(a+b\right)-2=0\Leftrightarrow a+b=...\)(Giải pt bậc 3 bằng máy tính)

\(A=\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}=\sqrt[3]{2^3+3.2^2.\sqrt{2}+3.2.\left(\sqrt{2}\right)^2+\left(\sqrt{2}\right)^3}+\sqrt[3]{2^3-3.2^2.\sqrt{2}+3.2.\left(\sqrt{2}\right)^2-\left(\sqrt{2}\right)^3}\)\(=\sqrt[3]{\left(2+\sqrt{2}\right)^3}+\sqrt[3]{\left(2-\sqrt{2}\right)^3}=2+\sqrt{2}+2-\sqrt{2}=4.\)

tự vẽ hình nha bn

a. Ta có: \(BC=\sqrt{AB^2+AC^2}=\sqrt{3^2+4^2}=5\)(Theo định lí Pytago, tam giác ABC vuông tại A)

b. Ta có: \(\frac{BH}{CH}=\frac{3}{4}\)

\(\Leftrightarrow\frac{BH+CH}{CH}=\frac{3}{4}+1\)

\(\Leftrightarrow\frac{BC}{CH}=\frac{7}{4}\)\(\Leftrightarrow\frac{5}{CH}=\frac{7}{4}\)\(\Leftrightarrow CH=\frac{5.4}{7}=\frac{20}{7}\)

\(\Rightarrow BH=5-\frac{20}{7}=\frac{15}{7}\)

c,d bạn giải giùm mình được không