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1.(x-y+z)2+(z-y)2+2(x-y+z)(y-z)= (x-y+z)+2(x-y+z)(y-z)+(y-z)2=(x-y+z+y-z)2=x2
CT : (A+B)2=A2+2AB+B2
Ta có : A = 4x - x2 + 3
=> A = -(x2 - 4x - 3)
=> A = -(x2 - 4x + 4 - 7)
=> A = -(x2 - 4x + 4) + 7
=> A = -(x - 2)2 + 7
Vì : \(-\left(x-2\right)^2\le0\forall x\)
=> A = -(x - 2)2 + 7 \(\le7\forall x\)
Vậy Amax = 7 khi x = 2
\(\left(x+y-z\right)^2+2\left(z-x-y\right)\left(x+y\right)+\left(x+y\right)^2\)
\(=\left(x+y-z\right)^2-2\left(x+y-z\right)\left(x+y\right)+\left(x+y\right)^2\)
\(\left[\left(x+y-z\right)-\left(x+y\right)\right]^2=z^2\)
\(a,\left(x+y\right)^2+\left(x-y\right)^2=x^2+2xy+y^2+x^2-2xy+y^2=2\left(x^2+y^2\right)\)\(b,2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2=2x^2-2y^2+x^2+2xy+y^2+x^2-2xy+y^2=3x^2\)\(c,\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)=\left[\left(x-y+z\right)-\left(z-y\right)\right]^2=\left(x-2y\right)^2\)
a) \(\left(x+y\right)^2+\left(x-y\right)^2\)
=\(\left(x^2+2xy+y^2\right)+\left(x^2-2xy+y^2\right)\)
=\(x^2+2xy+y^2+x^2-2xy+y^2\)
\(2x^2+2y^2=2\left(x^2+y^2\right)\)
b) \(2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2\)
\(=\left(x-y\right)^2+2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2\)
=\(\left[\left(x-y\right)+\left(x+y\right)\right]^2\)
= \(\left(x-y+x+y\right)^2\)
\(=2x^2\)
c) \(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2-2\left(x-y+z\right)\left(z-y\right)+\left(z-y\right)^2\)
\(=\left[\left(x-y+z\right)-\left(z-y\right)\right]^2\)
= \(\left(x-y+z-z+y\right)^2=x^2\)
Bài làm:
Ta có: \(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+2\left(x-y+z\right)\left(y-z\right)+\left(y-z\right)^2\)(hằng đẳng thức đầu)
\(=\left(x-y+z+y-z\right)^2=x^2\)
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+2\left(x-y+z\right)\left(y-z\right)+\left(y-z\right)^2\)
\(=\left[\left(x-y+z\right)+\left(y-z\right)\right]^2=\left(x-y+z+y-z\right)^2=x^2\)
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+\left(z-y\right)^2-2\left(x-y+z\right)\left(z-y\right)\)
\(=\left(x-y+z-z+y\right)^2\)
\(=x^2\)
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2-2\left(x-y+z\right)\left(z-y\right)+\left(z-y\right)^2\)
\(=\left(x-y+z-z+y\right)^2=x^2\)
Áp dụng hằng đẳng thức số 2 đó bạn!
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