\(a,P=\left(x-1\right)^2+4\ge4\\ P_{min}=4\Leftrightarrow x=1\\ b,Q=2\left(x^2-3x\right)=2\left(x^2-2\cdot\dfrac{3}{2}x+\dfrac{9}{4}\right)-\dfrac{9}{2}\\ Q=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\ge\dfrac{9}{2}\\ Q_{min}=\dfrac{9}{2}\Leftrightarrow x=\dfrac{3}{2}\\ c,M=\left(x^2-x+\dfrac{1}{4}\right)+\left(y^2+6y+9\right)+\dfrac{3}{4}\\ M=\left(x-\dfrac{1}{2}\right)^2+\left(y+3\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\\ M_{min}=\dfrac{3}{4}\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=-3\end{matrix}\right.\)
a) \(P=\left(x^2-2x+1\right)+4=\left(x-1\right)^2+4\ge4\)
\(minP=4\Leftrightarrow x=1\)
b) \(Q=2x^2-6x=2\left(x^2-3x+\dfrac{9}{4}\right)-\dfrac{9}{2}=2\left(x-\dfrac{3}{2}\right)^2-\dfrac{9}{2}\ge-\dfrac{9}{2}\)
\(minQ=-\dfrac{9}{2}\Leftrightarrow x=\dfrac{3}{2}\)
c) \(M=x^2+y^2-x+6y+10=\left(x^2-x+\dfrac{1}{4}\right)+\left(y^2+6y+9\right)+\dfrac{3}{4}=\left(x-\dfrac{1}{2}\right)^2+\left(y+3\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)
\(minM=\dfrac{3}{4}\Leftrightarrow\) \(\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=-3\end{matrix}\right.\)
