1: a) Ta có: \(A=x\left(x+2\right)+y\left(y-2\right)-2xy+37\)

\(=x^2+2x+y^2-2y-2xy+37\)

\(=\left(x^2-2xy+y^2\right)+\left(2x-2y\right)+37\)

\(=\left(x-y\right)^2+2\left(x-y\right)+37\)

\(=7^2+2.7+37\) (Vì \(x-y=7\))

\(=100\)

Vậy \(A=100\)

b) Ta có: \(B=x^2+4y^2-2x+10+4xy-4y\)

\(=\left(x^2+4xy+4y^2\right)-\left(2x+4y\right)+10\)

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

\(=5^2-2.5+10\)

\(=25\)

Vậy \(B=25\)

c) Ta có : \(C=\left(x-y\right)^2\)

\(=x^2-2xy+y^2\)

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

\(=26-2.5\) (Vì \(x^2+y^2=26\) ; \(xy=5\))

\(=16\)

Vậy \(C=16\)

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

\(=x^2+2xy\)

\(=x\left(x+2y\right)\) \(\left(dpcm\right)\)

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

\(=\left(x-y\right)^2\left(x+y\right)^2\) \(\left(dpcm\right)\)

c) \(\left(x+y\right)^2=x^2+2xy+y^2\)

\(=\left(x^2-2xy+y^2\right)+4xy\)

\(=\left(x-y\right)^2+4xy\) \(\left(dpcm\right)\)

Chúc bn học tốt ✔✔✔

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

b) \(\left(a+b\right)^2-\left(a-b\right)\left(a+b\right)=\left(a+b\right)^2-\left(a^2-b^2\right)=a^2+2ab+b^2-a^2+b^2\)

\(=2ab+2b^2=2b\left(a+b\right)\)

c)\(\left(a+b\right)^2-\left(a-b\right)^2=\left(a+b-a+b\right)\left(a+b+a-b\right)\)

\(=2b.2a=4ab\) 

a: \(\left(x+y\right)^2-2xy\)

\(=x^2+2xy+y^2-2xy\)

\(=x^2+y^2\)

b: \(\left(a+b\right)^2-\left(a-b\right)\left(a+b\right)\)

\(=\left(a+b\right)\left(a+b-a+b\right)\)

\(=2b\left(a+b\right)\)

c: \(\left(a+b\right)^2-\left(a-b\right)^2\)

\(=\left(a+b-a+b\right)\left(a+b+a-b\right)\)

\(=4ab\)

bài 2: a bạn có thể thêm bớt y^2 vào vế bên phải

bài 2 c thì bạn có thể mở ngoặc ở vế phải rồi tính sau đó áp dụng hđt

a)

\(\begin{array}{l}\left( {2x - 5y} \right)\left( {2x + 5y} \right) + {\left( {2x + 5y} \right)^2}\\ = \left( {2x + 5y} \right)\left( {2x - 5y + 2x + 5y} \right)\\ = \left( {2x + 5y} \right).4x\\ = 2x.4x + 5y.4x\\ = 8{x^2} + 20xy\end{array}\)

b)

\(\begin{array}{l}\left( {x + 2y} \right)\left( {{x^2} - 2xy + 4{y^2}} \right) + \left( {2x - y} \right)\left( {4{x^2} + 2xy + {y^2}} \right)\\ = {x^3} + {\left( {2y} \right)^3} + {\left( {2x} \right)^3} - {y^3}\\ = {x^3} + 8{y^3} + 8{x^3} - {y^3}\\ = \left( {{x^3} + 8{x^3}} \right) + \left( {8{y^3} - {y^3}} \right)\\ = 9{x^3} + 7{y^3}\end{array}\)

b: Ta có: \(\left(4x-y\right)\left(4x+y\right)-2\left(3x-2y\right)^2+\left(x-3y\right)^2\)

\(=16x^2-y^2-2\left(9x^2-12xy+4y^2\right)+x^2-6xy+9y^2\)

\(=17x^2-6xy+8y^2-18x^2+24xy-8y^2\)

\(=-x^2+18xy\)

c: Ta có: \(\left(2a-3b+4c\right)\left(2a-3b-4c\right)\)

\(=\left(2a-3b\right)^2-16c^2\)

\(=4a^2-12ab+9b^2-16c^2\)

1.a, VT= \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\)\(\left(x^2+y^2-2xy\right)\left(x^2+y^2+2xy\right)=\left(x-y\right)^2\left(x+y\right)^2=VP.\left(đpcm\right)\)

b, VP=\(x\left(x-3y\right)^2+y\left(y-3x\right)^2\)\(=x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\)\(=x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\)

\(=x^3+3x^2y+3xy^2+y^3\)\(=\left(x+y\right)^3=VT\left(đpcm\right)\)

2. VT=\(\left(a+b\right)^3-\left(a-b\right)^3\)\(=\left(a+b-a+b\right)\left(a^2+2ab+b^2+a^2-b^2+a^2-2ab+b^2\right)\)

\(2b\left(b^2+3a^2\right)\)\(=VP\left(đpcm\right)\).

a) (x² + y²)² - (2xy)²

= [(x² + y²) - 2xy].[(x² + y²) + 2xy]

= [x² + y² - 2xy].[(x² + y² + 2xy]

= (x - y)² . (x + y)²

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

Ta có : \(\left(x^2+y^2\right)^2-\left(2xy\right)^2=\left(x^2+y^2+2xy\right)\left(x^2+y^2-2xy\right)\)

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

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

ta có: \(x\left(x-3y\right)^2+y\left(y-3x\right)^2\)

\(=x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\)

\(=x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\)

= \(x^3+3x^2y+3xy^2+y^3\)

\(=\left(x+y\right)^3\)

2. \(\left(a+b\right)^3-\left(a-b\right)^3=2b\left(b^2+3a^2\right)\)

Ta có: \(\left(a+b\right)^3-\left(a-b\right)^3=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3\)

= \(2b^3+6a^2b\)

\(=2b\left(b^2+3a^2\right)\)

a. -(b-a)³= -b³+a³ (phá ngoặc trước có dấu trừ nên đổi dấu)

= a³ - b³ = (a-b)³

b)

\(\left(-a-b\right)^2=\left(-a\right)^2-2.\left(-a\right)b+b^2\\ =a^2+2ab+b^2=\left(a+b\right)^2\)

c) \(x\left(x-3y\right)^2+y\left(y-3x\right)^2\\ =x\left(x^2-6xy+9y^2\right)+y\left(y^2-6xy+9x^2\right)\\ =x^3-6x^2y+9xy^2+y^3-6xy^2+9x^2y\\ =x^3+3x^2y+3xy^2+y^3\\ =\left(x+y\right)^3\)

a. \(\left(20x^4y-25x^2y^2-3x^2y\right):5x^2y\)

\(=4x^2-5y-\frac{3}{5}\)

b. \(\left(15xy^2+17xy^3+18y^2\right):6y^2\)

\(=\frac{5}{2}x+\frac{17}{6}xy+3\)

c. \(\left[3\left(x-y\right)^4+2\left(x-y\right)^3-5\left(x-y\right)^2\right]:\left(y-x\right)^2\)

\(=\left[3\left(x-y\right)^4+2\left(x-y\right)^3-5\left(x-y\right)^2\right]:\left(x-y\right)^2\)

\(=3\left(x-y\right)^2+2\left(x-y\right)-5\)

d. \(\left(x^2-2xy+y^2\right):\left(y-x\right)\)

\(=\left(x-y\right)^2:\left(y-x\right)\)

\(=\left(y-x\right)^2:\left(y-x\right)\)

\(=y-x\)