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Bài 1 :
(a^2+b^2)(x^2+y^2)=(ax+by)^2
<=> a^2x^2 + a^2y^2 + b^2x^2 + b^2y^2 = a^2x^2 + 2abxy + b^2y^2
<=> a^2y^2 + b^2x^2 = 2abxy
<=> a^2y^2 + b^2x^2 - 2abxy = 0
<=> (ay - bx)^2 = 0
=> ay - bx = 0
=> ay = bx
=> a/x = b/y ( x,y khác 0)
Bài 2:
\(A=x^2+2xy+y^2-4x-4y+1\)
\(=\left(x+y\right)^2-4\left(x+y\right)+1\)
\(=3^2-4\cdot3+1=9-12+1=-2\)
\(\left(a+b+c+d\right)^2+\left(a+b-c-d\right)^2+\left(a-b+c-d\right)^2+\left(a-b-c+d\right)^2\)(Sửa lại nha bn viết sai để)
Đặt x=a+b , y=c+d , z=a-b , t=c-d
Khi đó biểu thức bằng
\(\left(x+y\right)^2+\left(x-y\right)^2+\left(z+t\right)^2+\left(z-t\right)^2\)
\(=x^2+y^2+2xy+x^2+y^2-2xy+z^2+t^2+2zt+z^2+t^2-2zt\)
\(=2\left(x^2+y^2+z^2+t^2\right)=2\left[\left(a+b\right)^2+\left(a-b\right)^2+\left(c+d\right)^2+\left(c-d\right)^2\right]\)
\(=2(a^2+b^2-2ab+a^2+b^2-2ab+c^2+d^2+2cd+c^2+d^2-2cd)\)
\(=2\left(2a^2+2b^2+2c^2+2d^2\right)=4\left(a^2+b^2+c^2+d^2\right)\)
Bài 1:
- a,(2+xy)^2=4+4xy+x^2y^2
- b,(5-3x)^2=25-30x+9x^2
- d,(5x-1)^3=125x^3 - 75x^2 + 15x^2 - 1
\(A=\left[\left(a+b\right)+\left(c+d\right)\right]^2+\left[\left(a+b\right)-\left(c+d\right)\right]^2+\left[\left(a-b\right)+\left(c-d\right)\right]^2+\left[\left(a-b\right)-\left(c-d\right)\right]^2\)
Ta có
\(\left[\left(a+b\right)+\left(c+d\right)\right]^2=\left(a+b\right)^2+2\left(a+b\right)\left(c+d\right)+\left(c+d\right)^2\)
\(\left[\left(a+b\right)-\left(c+d\right)\right]^2=\left(a+b\right)^2-2\left(a+b\right)\left(c+d\right)+\left(c+d\right)^2\)
\(\left[\left(a-b\right)+\left(c-d\right)\right]^2=\left(a-b\right)^2+2\left(a-b\right)\left(c-d\right)+\left(c-d\right)^2\)
\(\left[\left(a-b\right)-\left(c-d\right)\right]^2=\left(a-b\right)^2-2\left(a-b\right)\left(c-d\right)+\left(c-d\right)^2\)
\(A=2\left(a+b\right)^2+2\left(a-b\right)^2+2\left(c+d\right)^2+2\left(c-d\right)^2\)
\(A=2\left(a^2+2ab+b^2+a^2-2ab+b^2+c^2+2cd+d^2+c^2-2cd+d^2\right)\)
\(A=4\left(a^2+b^2+c^2+d^2\right)\)
1.
a) \((a + b + c)^2 + (a - b - c)^2 +( b - c - a) ^2 + (c - a - b)^2 \)
\(= (a + b + c)^2 + (a + b - c)^2 + (a - b - c)^2 + (a - b + c)^2 \)
\(= (a + b)^2 + 2c(a + b) + c^2 + (a + b)^2 - 2c(a + b) + c^2 + (a - b)^2 - 2c(a - b) + c^2 + (a - b)^2 + 2c(a - b) +c^2 \)
\(= 2(a + b)^2 + 2c^2 + 2(a - b)^2 + 2c^2 \)
\(= 2[(a + b)^2 + (a - b)^2] + 4c^2 \)
\(=2(2a^2 + 2b^2) + 4c^2 \)
\(= 4(a^2 + b^2 + c^2)\)
b) Đặt: \(x=a+b; y=c+d; z=a-b; t=c-d \)
Ta được:
\((x+y)^2+(x-y)^2+(z+t)^2+(z-t)^2 \)
\(= (x^2+2xy+y^2)+(x^2-2xy+y^2)+(z^2+2zt+t^2)+(z^2-2zt+t^2) \)
\(= 2x^2+2y^2+2z^2+2t^2 \)
\(= 2(x^2+y^2+z^2+t^2) \)
\(=2.\left[(a+b)^2+(c+d)^2+(a-b)^2+(c-d)^2 \right]\)
\(= 2(a^2+2ab+b^2+c^2+2cd+d^2+a^2-2ab+b^2+c^2-2cd+d^2) \)
\(= 2(2a^2+2b^2+2c^2+2d^2) \)
\(= 4(a^2+b^2+c^2+d^2)\)