数学 – 数学の威力を発揮する - 方程式 – 高次方程式(連立2次方程式, 2次と2次の場合, 適当な工夫)

数学読本〈1〉
数・方程式/方程式/不等式
(岩波書店)
松坂和夫 (著)

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数学読本〈1〉数・方程式/方程式/不等式 (松坂和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.3(高次方程式)、連立2次方程式の問36.を解いてみる。

問36.

$\begin{array}{l} (x - 3 y) (x - 2 y) = 0 \\ x = 3 y \\ 9 y^{2} + 3 y^{2} + 2 y^{2} = 56 \\ 14 y^{2} = 56 \\ y^{2} = 4 \\ y = \pm 2 \\ x = \pm 6, y = \pm 2 \\ x = 2 y \\ 4 y^{2} + 2 y^{2} + 2 y^{2} = 56 \\ 8 y^{2} = 56 \\ y^{2} = 7 \\ y = \pm \sqrt{7} \\ x = \pm 2 \sqrt{7}, y = \pm \sqrt{7} \end{array}$
$\begin{array}{l} 6 x^{2} + x y - 2 y^{2} = 0 \\ (3 x + 2 y) (2 x - y) = 0 \\ y = - \frac{3}{2} x \\ 5 x^{2} - \frac{9}{2} x^{2} = - 3 \\ x^{2} = - 6 \\ x = \pm \sqrt{6} i, y = \mp \frac{3 \sqrt{6} i}{2} \\ y = 2 x \\ 5 x^{2} - 8 x^{2} = - 3 \\ x^{2} = 1 \\ x = \pm 1, y = \pm 2 \end{array}$
$\begin{array}{l} y = - \frac{6}{x} \\ x^{2} + \frac{36}{x^{2}} = 13 \\ x^{4} - 13 x^{2} + 36 = 0 \\ (x^{2} - 4) (x^{2} - 9) = 0 \\ x^{2} = 4 \\ x = \pm 2, y = \mp 3 \\ x^{2} = 9 \\ x = \pm 3, y = \mp 2 \end{array}$
$\begin{array}{l} y = - \frac{15}{x} \\ x^{2} - \frac{225}{x^{2}} = 16 \\ x^{4} - 16 x^{2} - 225 = 0 \\ (x^{2} - 25) (x^{2} + 9) = 0 \\ x^{2} = 25 \\ x = \pm 5, y = \mp 3 \\ x^{2} = - 9 \\ x = \pm 3 i, y = \pm 5 i \end{array}$
$\begin{array}{l} x y = 11 - (x + y) \\ 22 - 2 (x + y) - (x + y) = 7 \\ x + y = 5 \\ x y = 6 \\ t^{2} - 5 t + 6 = 0 \\ (t - 3) (t - 2) = 0 \\ x = 3, y = 2 \\ x = 2, y = 3 \end{array}$
$\begin{array}{l} 60 x y + 48 x - 36 y - 24 = 0 \\ 60 x y + 45 - 35 y - 20 = 0 \\ 3 x - y - 4 = 0 \\ y = 3 x - 4 \\ 5 x (3 x - 4) + 4 x - 3 (3 x - 4) - 2 = 0 \\ 15 x^{2} - 25 x + 10 = 0 \\ 3 x^{2} - 5 x + 2 = 0 \\ (x - 1) (3 x - 2) = 0 \\ x = 1, y = - 1 \\ x = \frac{2}{3}, y = - 2 \end{array}$
$\begin{array}{l} (x + y) (x^{2} - x y + y^{2}) = - 4 \\ 2 (x^{2} - x y + y^{2}) = - 4 \\ x^{2} - x y + y^{2} = - 2 \\ y = 2 - x \\ x^{2} - 2 x + x^{2} + 4 + x^{2} - 4 x + 2 = 0 \\ 3 x^{2} - 6 x + 6 = 0 \\ x^{2} - 2 x + 2 = 0 \\ x = 1 \pm i, y = 1 \mp i \end{array}$
$\begin{array}{l} (x^{2} + y^{2}) (x^{2} - y^{2}) = - 65 \\ 13 (x^{2} + y^{2}) = - 65 \\ 13 x^{2} + 13 y^{2} + 65 = 0 \\ y^{2} = x^{2} - 13 \\ 13 x^{2} + 13 x^{2} - 169 + 65 = 0 \\ 26 x^{2} = 104 \\ x^{2} = 4 \\ x = \pm 2, y = \pm 3 i \\ x = \mp 2, y = \pm 3 i \end{array}$

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2015年5月21日木曜日

数学 – 数学の威力を発揮する - 方程式 – 高次方程式(連立2次方程式, 2次と2次の場合, 適当な工夫)

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