与えられた数学の問題は、以下の20個です。 (1) $\sqrt{28} \div \sqrt{2}$ (2) $2a(a-2) + 3a(1+3a)$ (3) $3\sqrt{6} \div \sqrt{2} \times 4\sqrt{3}$ (4) $5\sqrt{5} - \sqrt{2} - 3\sqrt{5} + 4\sqrt{2}$ (5) $\sqrt{35} \times 2\sqrt{7}$ (6) $4a(a-2b-2)$ (7) $\sqrt{28} - \sqrt{7}$ (8) $(6x^2 - 9x) \div (-3x)$ (9) $(x+3)(x+7)$ (10) $(4a-8b)(4a+8b)$ (11) $(x-3)^2$ (12) $(x+3)^2 + (x-12)(x+3)$ (13) $2x^2 + 14x + 20$ (14) $6x^2 - 9$ (15) $a^2 + 12a + 36$ (16) $x^2 + 5x + 6$ (17) $x^2 + 8x + 12 = 0$ (18) $x^2 - 6x + 7 = 0$ (19) $(x+5)(x+3) = 2(x^2+3)$ (20) $x^2 = x$

はい、承知いたしました。与えられた問題を解いていきます。

1. 問題の内容

与えられた数学の問題は、以下の20個です。

(1) $\sqrt{28} \div \sqrt{2}$

(2) $2a(a-2) + 3a(1+3a)$

(3) $3\sqrt{6} \div \sqrt{2} \times 4\sqrt{3}$

(4) $5\sqrt{5} - \sqrt{2} - 3\sqrt{5} + 4\sqrt{2}$

(5) $\sqrt{35} \times 2\sqrt{7}$

(6) $4a(a-2b-2)$

(7) $\sqrt{28} - \sqrt{7}$

(8) $(6x^2 - 9x) \div (-3x)$

(9) $(x+3)(x+7)$

(10) $(4a-8b)(4a+8b)$

(11) $(x-3)^2$

(12) $(x+3)^2 + (x-12)(x+3)$

(13) $2x^2 + 14x + 20$

(14) $6x^2 - 9$

(15) $a^2 + 12a + 36$

(16) $x^2 + 5x + 6$

(17) $x^2 + 8x + 12 = 0$

(18) $x^2 - 6x + 7 = 0$

(19) $(x+5)(x+3) = 2(x^2+3)$

(20) $x^2 = x$

2. 解き方の手順

(1) $\sqrt{28} \div \sqrt{2} = \sqrt{\frac{28}{2}} = \sqrt{14}$

(2) $2a(a-2) + 3a(1+3a) = 2a^2 - 4a + 3a + 9a^2 = 11a^2 - a$

(3) $3\sqrt{6} \div \sqrt{2} \times 4\sqrt{3} = 3 \times \frac{\sqrt{6}}{\sqrt{2}} \times 4\sqrt{3} = 3 \times \sqrt{3} \times 4\sqrt{3} = 12 \times 3 = 36$

(4) $5\sqrt{5} - \sqrt{2} - 3\sqrt{5} + 4\sqrt{2} = (5-3)\sqrt{5} + (-1+4)\sqrt{2} = 2\sqrt{5} + 3\sqrt{2}$

(5) $\sqrt{35} \times 2\sqrt{7} = 2\sqrt{5 \times 7} \times \sqrt{7} = 2\sqrt{5} \times \sqrt{7} \times \sqrt{7} = 2 \times 7 \times \sqrt{5} = 14\sqrt{5}$

(6) $4a(a-2b-2) = 4a^2 - 8ab - 8a$

(7) $\sqrt{28} - \sqrt{7} = \sqrt{4 \times 7} - \sqrt{7} = 2\sqrt{7} - \sqrt{7} = \sqrt{7}$

(8) $(6x^2 - 9x) \div (-3x) = \frac{6x^2 - 9x}{-3x} = \frac{6x^2}{-3x} - \frac{9x}{-3x} = -2x + 3$

(9) $(x+3)(x+7) = x^2 + 7x + 3x + 21 = x^2 + 10x + 21$

(10) $(4a-8b)(4a+8b) = (4a)^2 - (8b)^2 = 16a^2 - 64b^2$

(11) $(x-3)^2 = x^2 - 2(3)x + 3^2 = x^2 - 6x + 9$

(12) $(x+3)^2 + (x-12)(x+3) = (x+3)(x+3 + x - 12) = (x+3)(2x - 9) = 2x^2 - 9x + 6x - 27 = 2x^2 - 3x - 27$

(13) $2x^2 + 14x + 20 = 2(x^2 + 7x + 10) = 2(x+2)(x+5)$

(14) $6x^2 - 9 = 3(2x^2 - 3) = 3(\sqrt{2}x - \sqrt{3})(\sqrt{2}x + \sqrt{3})$

(15) $a^2 + 12a + 36 = (a+6)^2$

(16) $x^2 + 5x + 6 = (x+2)(x+3)$

(17) $x^2 + 8x + 12 = 0$

$(x+6)(x+2) = 0$

$x = -6, -2$

(18) $x^2 - 6x + 7 = 0$

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

$x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(7)}}{2(1)} = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2} = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}$

(19) $(x+5)(x+3) = 2(x^2+3)$

$x^2 + 8x + 15 = 2x^2 + 6$

$x^2 - 8x - 9 = 0$

$(x-9)(x+1) = 0$

$x = 9, -1$

(20) $x^2 = x$

$x^2 - x = 0$

$x(x-1) = 0$

$x = 0, 1$

3. 最終的な答え

(1) $\sqrt{14}$

(2) $11a^2 - a$

(3) $36$

(4) $2\sqrt{5} + 3\sqrt{2}$

(5) $14\sqrt{5}$

(6) $4a^2 - 8ab - 8a$

(7) $\sqrt{7}$

(8) $-2x + 3$

(9) $x^2 + 10x + 21$

(10) $16a^2 - 64b^2$

(11) $x^2 - 6x + 9$

(12) $2x^2 - 3x - 27$

(13) $2(x+2)(x+5)$

(14) $3(\sqrt{2}x - \sqrt{3})(\sqrt{2}x + \sqrt{3})$

(15) $(a+6)^2$

(16) $(x+2)(x+3)$

(17) $x = -6, -2$

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

(19) $x = 9, -1$

(20) $x = 0, 1$