a: \(A=x^2-8x+16+4\)
\(=\left(x-4\right)^2+4>0\forall x\)
b \(B=4x^2-12x+9+2\)
\(=\left(2x-3\right)^2+2>0\forall x\)
a) \(A=\left(x^2-8x+16\right)+4=\left(x-4\right)^2+4\ge4>0\)
b) \(B=\left(4x^2-12x+9\right)+2=\left(2x-3\right)^2+2\ge2>0\)
c) \(C=\left(x^2-3x+\dfrac{9}{4}\right)+\dfrac{11}{4}=\left(x-\dfrac{3}{2}\right)^2+\dfrac{11}{4}\ge\dfrac{11}{4}>0\)
d) \(D=\left(x^2+2x+1\right)+5\left(y^2+\dfrac{6}{5}y+\dfrac{9}{25}\right)+\dfrac{156}{5}\)
\(=\left(x+1\right)^2+5\left(y+\dfrac{3}{5}\right)^2+\dfrac{156}{5}\ge\dfrac{156}{5}>0\)
e) \(E=\left(x^2-2x+1\right)+\left(y^2+4y+4\right)+1\)
\(=\left(x-1\right)^2+\left(y+4\right)^2+1\ge1>0\)
g) \(G=225x^2-30x+1+21x^2+30x+9-x^2+73\)
\(=245x^2+83\ge83>0\)
k) \(K=\left[y^2-2y\left(2x+1\right)+\left(2x+1\right)^2\right]+\left(x^2+4x+4\right)+2015\)
\(=\left(y-2x-1\right)^2+\left(x+2\right)^2+2015\ge2015>0\)
