Bài 10:
a: \(x^2+2xy+y^2+1\)
\(=\left(x^2+2xy+y^2\right)+1\)
\(=\left(x+y\right)^2+1\ge1>0\forall x,y\)
b: \(x^2+y^2+1\ge xy+x+y\)
=>\(2x^2+2y^2+2\ge2xy+2x+2y\)
=>\(2x^2+2y^2+2-2xy-2x-2y\ge0\)
=>\(\left(x^2-2xy+y^2\right)+\left(x^2-2x+1\right)+\left(y^2-2y+1\right)\ge0\)
=>\(\left(x-y\right)^2+\left(x-1\right)^2+\left(y-1\right)^2\ge0\forall x,y\) (luôn đúng)
c: \(x^2-x+1=x^2-x+\frac14+\frac34\)
\(=x^2-2\cdot x\cdot\frac12+\left(\frac12\right)^2+\frac34\)
\(=\left(x-\frac12\right)^2+\frac34\ge\frac34>0\forall x\)
Bài 8:
a: \(A=5x-x^2\)
\(=-\left(x^2-5x\right)\)
\(=-\left(x^2-5x+\frac{25}{4}-\frac{25}{4}\right)\)
\(=-\left(x-\frac52\right)^2+\frac{25}{4}\le\frac{25}{4}\forall x\)
Dấu '=' xảy ra khi \(x-\frac52=0\)
=>\(x=\frac52\)
b: \(B=x-x^2\)
\(=-x^2+x-\frac14+\frac14\)
\(=-\left(x^2-x+\frac14\right)+\frac14=-\left(x-\frac12\right)^2+\frac14\le\frac14\forall x\)
Dấu '=' xảy ra khi \(x-\frac12=0\)
=>\(x=\frac12\)
c: \(C=4x-x^2+3\)
\(=-\left(x^2-4x-3\right)\)
\(=-\left(x^2-4x+4-7\right)=-\left(x-2\right)^2+7\le7\forall x\)
Dấu '=' xảy ra khi x-2=0
=>x=2
Bài 6:
a: \(892^2+892\cdot216+108^2\)
\(=892^2+2\cdot892\cdot108+108^2\)
\(=\left(892+108\right)^2=1000^2=1000000\)
b: \(10,2\cdot9,8-9,8\cdot0,2+10,2^2-10,2\cdot0,2\)
\(=9,8\cdot\left(10,2-0,2\right)+10,2\left(10,2-0,2\right)\)
\(=10\cdot\left(9,8+10,2\right)=10\cdot20=200\)
