与えられた二次方程式を解きます。

1. 問題の内容

与えられた二次方程式を解きます。

2. 解き方の手順

(3) $8x^2 = 2$

$x^2 = \frac{2}{8} = \frac{1}{4}$

$x = \pm \sqrt{\frac{1}{4}}$

$x = \pm \frac{1}{2}$

(4) $3x^2 - 12 = 0$

$3x^2 = 12$

$x^2 = \frac{12}{3} = 4$

$x = \pm \sqrt{4}$

$x = \pm 2$

(5) $4x^2 - 9 = 0$

$4x^2 = 9$

$x^2 = \frac{9}{4}$

$x = \pm \sqrt{\frac{9}{4}}$

$x = \pm \frac{3}{2}$

(6) $5x^2 - 60 = 0$

$5x^2 = 60$

$x^2 = \frac{60}{5} = 12$

$x = \pm \sqrt{12}$

$x = \pm 2\sqrt{3}$

(7) $(x - 4)^2 = 24$

$x - 4 = \pm \sqrt{24}$

$x - 4 = \pm 2\sqrt{6}$

$x = 4 \pm 2\sqrt{6}$

(8) $(x + 1)^2 = 2$

$x + 1 = \pm \sqrt{2}$

$x = -1 \pm \sqrt{2}$

(9) $(x - 1)^2 - 8 = 0$

$(x - 1)^2 = 8$

$x - 1 = \pm \sqrt{8}$

$x - 1 = \pm 2\sqrt{2}$

$x = 1 \pm 2\sqrt{2}$

(10) $(x + 4)^2 - 10 = 8$

$(x + 4)^2 = 18$

$x + 4 = \pm \sqrt{18}$

$x + 4 = \pm 3\sqrt{2}$

$x = -4 \pm 3\sqrt{2}$

(11) $(x - 5)^2 - 4 = 5$

$(x - 5)^2 = 9$

$x - 5 = \pm \sqrt{9}$

$x - 5 = \pm 3$

$x = 5 \pm 3$

$x = 8, 2$

(12) $(x + 2)^2 - 36 = 0$

$(x + 2)^2 = 36$

$x + 2 = \pm \sqrt{36}$

$x + 2 = \pm 6$

$x = -2 \pm 6$

$x = 4, -8$

(13) $x^2 + 4x - 1 = 0$

解の公式: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-4 \pm \sqrt{16 + 4}}{2}$

$x = \frac{-4 \pm \sqrt{20}}{2}$

$x = \frac{-4 \pm 2\sqrt{5}}{2}$

$x = -2 \pm \sqrt{5}$

(14) $4x^2 + 3x - 2 = 0$

解の公式: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-2)}}{2(4)}$

$x = \frac{-3 \pm \sqrt{9 + 32}}{8}$

$x = \frac{-3 \pm \sqrt{41}}{8}$

(15) $x^2 + 11x - 1 = 0$

解の公式: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$x = \frac{-11 \pm \sqrt{11^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-11 \pm \sqrt{121 + 4}}{2}$

$x = \frac{-11 \pm \sqrt{125}}{2}$

$x = \frac{-11 \pm 5\sqrt{5}}{2}$

(16) $2x^2 + 3x + 1 = 0$

因数分解: $(2x+1)(x+1) = 0$

$2x + 1 = 0$ or $x + 1 = 0$

$2x = -1$ or $x = -1$

$x = -\frac{1}{2}$ or $x = -1$

3. 最終的な答え

(3) $x = \pm \frac{1}{2}$

(4) $x = \pm 2$

(5) $x = \pm \frac{3}{2}$

(6) $x = \pm 2\sqrt{3}$

(7) $x = 4 \pm 2\sqrt{6}$

(8) $x = -1 \pm \sqrt{2}$

(9) $x = 1 \pm 2\sqrt{2}$

(10) $x = -4 \pm 3\sqrt{2}$

(11) $x = 8, 2$

(12) $x = 4, -8$

(13) $x = -2 \pm \sqrt{5}$

(14) $x = \frac{-3 \pm \sqrt{41}}{8}$

(15) $x = \frac{-11 \pm 5\sqrt{5}}{2}$

(16) $x = -\frac{1}{2}, -1$