【数学I】例題3.3.9：放物線とx軸の共有点の座標（One More）★

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% 例題I3.3.9：放物線と$x$軸との共有点の座標（One More）★
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次の放物線は$x$軸と共有点をもつか．もつときは，その座標を求めよ．

(1) $y=-x^2+6x-9$

(2) $y=x^2+2x-8$

(3) $y=4x^2-5x+5$

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% 例題I3.3.9の解答（One More）★
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(1) $-x^2+6x-9=0$とすると，$x^2-6x+9=0$

したがって，$(x-3)^2=0$

ゆえに，$x=3$

よって，$x$軸と共有点を1個もち，その座標は，$(3,0)$

(2) $x^2+2x-8=0$とすると，$(x-2)(x+4)=0$

したがって，$x=2,-4$

よって，$x$軸と共有点を2個もち，その座標は，$(2,0),(-4,0)$

(3) $4x^2-5x+5=0$とする．この2次方程式の判別式を$D$とすると，

\[
D=(-5)^2-4\cdot 4\cdot 5=-55
\]

よって，$D<0$であるから，グラフは$x$軸と共有点をもたない．

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% 例題I3.3.9の別解
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\[
y=4x^2-5x+5=4\{(x-\frac{5}{8})^2-(\frac{5}{8})^2\}+5=4(x-\frac{5}{8})^2+\frac{55}{16}
\]

2次関数のグラフは，下に凸の放物線で，頂点の$y$座標は$\frac{55}{16}$である．

よって，グラフは$x$軸と共有点をもたない．

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% 問題I3.3.9：放物線と$x$軸との共有点の座標（One More）★
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次の放物線は$x$軸と共有点をもつか．もつときは，その座標を求めよ．

(1) $y=x^2-3x-10$

(2) $y=-x^2+4x-4$

(3) $y=3x^2-2x+6$

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% 問題I3.3.9の解答（One More）★
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(1) $x^2-3x-10=0$とすると，$(x-5)(x+2)=0$

したがって，$x=5,-2$

よって，$x$軸と共有点を2個もち，その座標は，$(5,0),(-2,0)$

(2) $-x^2+4x-4=0$とすると，$x^2-4x+4=0$

したがって，$(x-2)^2=0$

ゆえに，$x=2$

よって，$x$軸と共有点を1個もち，その座標は，$(2,0)$

(3) $3x^2-2x+6=0$とする．この2次方程式の判別式を$D$とすると，

\[
D=(-2)^2-4\cdot 3\cdot 6=-68
\]

よって，$D<0$であるから，グラフは$x$軸と共有点をもたない．

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% 問題I3.3.9の別解
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\[
y=3x^2-2x+6=3\{(x-\frac{1}{3})^2-\frac{1}{9}\}+6=3(x-\frac{1}{3})^2+\frac{17}{3}
\]

2次関数のグラフは，下に凸の放物線で，頂点の$y$座標は$\frac{17}{3}$である．

よって，グラフは$x$軸と共有点をもたない．

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% 例題I3.3.9の側注：放物線と$x$軸との共有点の座標（One More）★
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% 側注1
判別式を$D$とすると，

\[
D=(-6)^2-4\cdot 1\cdot 9=0
\]

よって，$D=0$より，共有点は1個（重解）である．また，$(3,0)$は接点となる．

% 側注2
判別式を$D$とすると，

\[
D=2^2-4\cdot 1\cdot(-8)=36
\]
よって，$D>0$より，共有点は2個である（$\frac{D}{4}$を用いてもよい）．

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% 問題I3.3.9の側注：放物線と$x$軸との共有点の座標（One More）★
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% 側注1
判別式を$D$とすると，

\[
D=(-3)^2-4\cdot 1\cdot(-10)=49
\]
よって，$D>0$より，共有点は2個である．

% 側注2
判別式を$D$とすると，

\[
D=4^2-4\cdot 1\cdot 4=0
\]

よって，$D=0$より，共有点は1個（重解）である．また，$(2,0)$は接点となる．

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% 例題I3.3.9の考え方・補足など：放物線と$x$軸との共有点の座標（One More）★
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% 考え方
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$y=ax^2+bx+c$のグラフと$x$軸との共有点の$x$座標は，2次関数の式において$y=0$とおいたときの2次方程式$ax^2+bx+c=0$の実数解である．

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% 放物線と$x$軸の共有点の個数
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2次関数を$y=0$とおいた2次方程式$ax^2+bx+c=0$の判別式を$D=b^2-4ac$とすると，

\begin{align*}
&D>0\Longleftrightarrow x\text{軸との共有点は2個（軸と異なる2点で交わる）}\\
&D=0\Longleftrightarrow x\text{軸との共有点は1個（軸に接する）}\\
&D<0\Longleftrightarrow x\text{軸との共有点は0個（軸と共有点をもたない）}
\end{align*}