第4章 4.13

(1)

$x^2 - 5 = (x+\sqrt{5})(x-\sqrt{5})$

(2)

$x^2+2x-5$

$x = -1\pm\sqrt{1+5} = -1\pm\sqrt{6}\quad\text{これから}$

$(x+1+\sqrt{6})(x+1-\sqrt{6})$

(3)

$a^2+a-1$

$a = \dfrac{-1\pm\sqrt{1+4}}{2} = \dfrac{-1\pm\sqrt{5}}{2}\quad\text{これから}$

$\left(x+\dfrac{1+\sqrt{5}}{2}\right)\left(x+\dfrac{1-\sqrt{5}}{2}\right)$

(4)

$x^2+2x-4$

$x = -1\pm\sqrt{1+4} = -1\pm\sqrt{5}\quad\text{これから}$

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

(5)

$4x^2+4x-1$

$x = \dfrac{-2\pm\sqrt{4+4}}{4} = \dfrac{-2\pm2\sqrt{2}}{4} = \dfrac{-1\pm\sqrt{2}}{2}\quad\text{これから}$

$\left(x+\dfrac{1+\sqrt{2}}{2}\right)\left(x+\dfrac{1-\sqrt{2}}{2}\right)$

(6)

$a^2 + 2ab - 5b^2$

$a = -b \pm \sqrt{b^2+5b^2} = -b \pm \sqrt{6}b$ これから

$(a+b+\sqrt{6}b)(a+b-\sqrt{6}b)$