H1

(1) \(y=x^2+6x+2\) $$\begin{align*} y &= x^2 + 6x + 2\\ &= (x+3)^2 - 3^2 + 2\\ &= (x+3)^2-7 \end{align*}$$ (2) \(y=x^2-5x\) $$\begin{align*} y &= x^2-5x\\ &= \left(x-\frac{5}{2}\right)^2-\left(\frac{5}{2}\right)^2\\ &= \left(x-\frac{5}{2}\right)^2-\frac{25}{4} \end{align*}$$ (3) \(y=3x^2-6+11\) $$\begin{align*} y &= 3x^2 - 6x + 11\\ &= 3(x^2 - 2x) + 11\\ &= 3(x-1)^2-3\times1^2+11 \\ &{\scriptsize \uparrow \text{補足の\(p=-1\)を使えばここから}}\\ &= 3(x-1)^2+8 \end{align*}$$ (4) \(y=-2x^2+7x-3\) $$\begin{align*} y &= -2x^2 + 7x - 3\\ &= -2\left(x^2-\frac{7}{2}x\right)-3\\ &= -2\left[\left(x-\frac{7}{4}\right)^2-\left(\frac{7}{4}\right)^2\right]-3\\ &= -2\left(x-\frac{7}{4}\right)^2+2\times\frac{49}{16}-3\\ &= -2\left(x-\frac{7}{4}\right)^2+\frac{25}{8} \end{align*}$$ (補足を使った場合)
(4) \(y = -2x^2 + 7x - 3\) $$\begin{align*} y &= -2x^2 + 7x - 3\\ &= -2\left[x+\frac{7}{2\times(-2)}\right]^2{\color{red}+}2\times\left[\frac{7}{2\times(-2)}\right]^2-3\\ &= -2\left(x-\frac{7}{4}\right)^2+\frac{49}{8}-3\\ &= -2\left(x-\frac{7}{4}\right)^2+\frac{25}{8} \end{align*}$$