`a, (x-y)^2 = (x+y)^2 - 4xy = 12^2 - 35 . 4 = 144 - 140 = 4`.

`b, (x+y)^2 = (x-y)^2 + 4xy = 8^2 + 20.4 = 64 + 80 = 144`

`c, x^3 + y^3 = (x+y)^3 - 3xy(x+y) = 5^3 - 3 . 6 . 5 = 125 - 90 = 35`

`d, x^3 - y^3 = (x-y)^3 - 3xy(x-y) = 3^3 - 3 .40 . 3 = 27 - 360 = -333`.

Ta có x + y = 3

=> (x + y)² = 9

<=> x²  + y² + 2xy = 9

<=> 2xy = 4

<=> xy = 2

Khi đó x³ + y³  = (x +  y)(x² - xy + y²) = 3.(5 - 2) = 9

b) Ta có x - y = 5 

<=> (x - y)² = 25

<=> x² - 2xy + y² = 25

<=> -2xy = 10

<=> xy = -5

Khi đó x³ - y³ = (x - y)(x² - xy + y²) = 5.(15 + 5) = 100 

Bài 2: 

Đặt \(\dfrac{x}{3}=\dfrac{y}{4}=k\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=3k\\y=4k\end{matrix}\right.\)

Ta có: xy=12

\(\Leftrightarrow12k^2=12\)

\(\Leftrightarrow k^2=1\)

Trường hợp 1: k=1

\(\Leftrightarrow\left\{{}\begin{matrix}x=3k=3\\y=4k=4\end{matrix}\right.\)

Trường hợp 2: k=-1

\(\Leftrightarrow\left\{{}\begin{matrix}x=3k=-3\\y=4k=-4\end{matrix}\right.\)

\(x-y=-30\Rightarrow\dfrac{x}{-30}=\dfrac{1}{y}\\ y.z=-42\\ \Rightarrow\dfrac{z}{-42}=\dfrac{1}{y}\\ \Rightarrow\dfrac{x}{-30}=\dfrac{z}{-42}\)

Áp dụng TCDTSBN ta có:

\(\dfrac{x}{-30}=\dfrac{z}{-42}=\dfrac{z-x}{-42-\left(-30\right)}=\dfrac{-12}{-12}=1\)

\(\dfrac{x}{-30}=1\Rightarrow x=-30\\ \dfrac{z}{-42}=1\Rightarrow z=-42\)

\(x.y=-30\Rightarrow-30.y=-30\Rightarrow y=1\)

\(P=\sum\dfrac{1}{x+y+1}\ge\dfrac{9}{2\left(x+y+z\right)+3}=\dfrac{9}{2.1+3}=\dfrac{9}{5}\)

Dấu \("="\Leftrightarrow x=y=z=\dfrac{1}{3}\)

A nhé

Chọn A

1) Cho x+y=2 và x^2+y^2=10. Tính x^3+y^3. Giải

(x+y)^2=x^2+y^2+2xy => xy= -3 
x^3+y^3=(x+y)^3-3xy(x+y) = 26

2) Ta có: x^3+y^3 = (x+y)(x^2-xy+y^2) (1)

(x+y)^2=a^2

=> x^2 +2xy +y^2=a^2

=> b+2xy=a^2

=> xy=\(\frac{a^2-b}{2}\)

Thay (1) vào đó ta có:

x^3+y^3= (x+y)(x^2-xy+y^2) = a(b-\(\frac{a^2-b}{2}\)) = \(a\left(\frac{2b-a^2+b}{2}\right)=a.\frac{3b-a^2}{2}\)

\(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=2\left(10-xy\right)\)

Ta có: \(x^2+y^2=\left(x+y\right)^2-2xy=2^2-2xy=4-2xy=10\Rightarrow2xy=-6\Rightarrow xy=-3\)

Vậy: \(x^3+y^3=2\left(10+3\right)=2.13=26\)

\(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=a\left(b-xy\right)\)

a) \(\dfrac{x}{y}=\dfrac{9}{11}\)

\(\Rightarrow\dfrac{x}{9}=\dfrac{y}{11}=\dfrac{x+y}{9+11}=\dfrac{60}{20}=3\)

\(\Rightarrow\left\{{}\begin{matrix}x=3.9=27\\y=3.11=33\end{matrix}\right.\)

b) \(7x=4y\Rightarrow\dfrac{x}{4}=\dfrac{y}{7}\)

\(\Rightarrow\dfrac{x}{4}=\dfrac{y}{7}=\dfrac{y-x}{7-4}=\dfrac{24}{3}=8\)

\(\Rightarrow\left\{{}\begin{matrix}x=8.4=32\\y=8.7=56\end{matrix}\right.\)

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