数学学習の記録 130 数・式の計算・方程式不等式 (数学読本) 第3章問11,12,13

"数・式の計算・方程式不等式 (数学読本)"の第3章問11,12,13を解いてみる。

問11

(1)

(2)

$\frac{1\pm\sqrt{5}}{2}$

$(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\sqrt{5}}{2})$

(3)

(4)

$\frac{2\pm\sqrt{4-9}}{3}\\ =\frac{2\pm\sqrt{5}i}{3}$

$3(x-\frac{2+\sqrt{5}i}{3})(x-\frac{2-\sqrt{5}i}{3})$

(5)

$\frac{-7\sqrt{2}\pm\sqrt{98-26}}{2}\\ =\frac{-7\sqrt{2}\pm6\sqrt{2}}{2}$

$=\frac{-\sqrt{2}}{2},\ \ \ -\frac{13\sqrt{2}}{2}$

$2(x+\frac{\sqrt{2}}{2})(x+\frac{13\sqrt{2}}{2})\\ =(\sqrt{2}x+1)(\sqrt{2}x+13)$

問12

$ax^{2}+bxy+cy^{2}\\ =a(x^{2}+\frac{b}{a}yx+\frac{c}{a}y^{2})\\ =a(x-(\alpha+\beta)yx+\alpha\beta y^{2})$

$=a(x-\alpha y)(x-\beta y)$

問13

実数の範囲

$(x^{2}+x+1)(x^{2}-x+1)$

複素数の範囲

$\frac{-1\pm\sqrt{3}i}{2},\ \ \ \frac{1\pm\sqrt{3}i}{2}$

$(x-\frac{-1+\sqrt{3}i}{2})(x+\frac{1+\sqrt{3}i}{2})(x-\frac{1+\sqrt{3}i}{2})(x-\frac{1-\sqrt{3}i}{2})$

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