与えられた数式を計算しなさい。具体的には、展開、因数分解、または簡略化を行う問題です。

1. 問題の内容

与えられた数式を計算しなさい。具体的には、展開、因数分解、または簡略化を行う問題です。

2. 解き方の手順

問題4 (1): $(x+7y)^2$

$(x+7y)^2 = x^2 + 2(x)(7y) + (7y)^2 = x^2 + 14xy + 49y^2$

問題4 (2): $(4x-y)^2$

$(4x-y)^2 = (4x)^2 - 2(4x)(y) + y^2 = 16x^2 - 8xy + y^2$

問題4 (3): $(5x+2y)^2$

$(5x+2y)^2 = (5x)^2 + 2(5x)(2y) + (2y)^2 = 25x^2 + 20xy + 4y^2$

問題4 (4): $(0.3a - 0.2b)^2$

$(0.3a - 0.2b)^2 = (0.3a)^2 - 2(0.3a)(0.2b) + (0.2b)^2 = 0.09a^2 - 0.12ab + 0.04b^2$

問題4 (5): $(-2a+5b)^2$

$(-2a+5b)^2 = (-2a)^2 + 2(-2a)(5b) + (5b)^2 = 4a^2 - 20ab + 25b^2$

問題4 (6): $(-3x - 2y)^2$

$(-3x - 2y)^2 = (-3x)^2 + 2(-3x)(-2y) + (-2y)^2 = 9x^2 + 12xy + 4y^2$

問題5 (1): $(x+11)(x-11)$

$(x+11)(x-11) = x^2 - 11^2 = x^2 - 121$

問題5 (2): $(a+8)(8-a)$

$(a+8)(8-a) = (8+a)(8-a) = 8^2 - a^2 = 64 - a^2$

問題5 (3): $(x-5y)(x+5y)$

$(x-5y)(x+5y) = x^2 - (5y)^2 = x^2 - 25y^2$

問題5 (4): $(9a-2b)(9a+2b)$

$(9a-2b)(9a+2b) = (9a)^2 - (2b)^2 = 81a^2 - 4b^2$

問題5 (5): $(a-0.1b)(0.1b+a)$

$(a-0.1b)(0.1b+a) = (a-0.1b)(a+0.1b) = a^2 - (0.1b)^2 = a^2 - 0.01b^2$

問題5 (6): $(-7x+6y)(7x+6y)$

$(-7x+6y)(7x+6y) = (6y-7x)(6y+7x) = (6y)^2 - (7x)^2 = 36y^2 - 49x^2$

問題6 (1): $(x+6)(x-2)-2(2x-1)$

$(x+6)(x-2)-2(2x-1) = x^2 + 4x - 12 - 4x + 2 = x^2 - 10$

問題6 (2): $(x-3)^2 + (x-2)(x+2)$

$(x-3)^2 + (x-2)(x+2) = x^2 - 6x + 9 + x^2 - 4 = 2x^2 - 6x + 5$

問題6 (3): $(x-4)^2 + (x-5)^2$

$(x-4)^2 + (x-5)^2 = x^2 - 8x + 16 + x^2 - 10x + 25 = 2x^2 - 18x + 41$

問題6 (4): $(a+4)^2 - (a-6)(a-2)$

$(a+4)^2 - (a-6)(a-2) = a^2 + 8a + 16 - (a^2 - 8a + 12) = a^2 + 8a + 16 - a^2 + 8a - 12 = 16a + 4$

問題6 (5): $(5x+1)(5x-1) - 2(x-5)^2$

$(5x+1)(5x-1) - 2(x-5)^2 = 25x^2 - 1 - 2(x^2 - 10x + 25) = 25x^2 - 1 - 2x^2 + 20x - 50 = 23x^2 + 20x - 51$

問題6 (6): $(x-7)(x+7) - (x-9)(x+9)$

$(x-7)(x+7) - (x-9)(x+9) = x^2 - 49 - (x^2 - 81) = x^2 - 49 - x^2 + 81 = 32$

問題7 (1): $(x-3y+5)^2$

$(x-3y+5)^2 = (x-3y+5)(x-3y+5) = x^2 + 9y^2 + 25 - 6xy + 10x - 30y$

問題7 (2): $(a+2b-5)(a+2b+4)$

$(a+2b-5)(a+2b+4) = ((a+2b) - 5)((a+2b)+4) = (a+2b)^2 - (a+2b) - 20 = a^2 + 4ab + 4b^2 + 2b*a -a-2b - 20 = a^2 + 4ab + 4b^2 -a - 2b - 20$

問題7 (3): $(3x+4y-6)(3x+4y+9)$

$(3x+4y-6)(3x+4y+9) = ((3x+4y)-6)((3x+4y)+9) = (3x+4y)^2 + 3(3x+4y) - 54 = 9x^2 + 24xy + 16y^2 + 9x + 12y - 54$

問題7 (4): $(3x-4y-6)(3x-4y+8)$

$(3x-4y-6)(3x-4y+8) = ((3x-4y)-6)((3x-4y)+8) = (3x-4y)^2 + 2(3x-4y) - 48 = 9x^2 -24xy + 16y^2 + 6x - 8y - 48$

問題7 (5): $(5a+3b-7)(5a+3b+7)$

$(5a+3b-7)(5a+3b+7) = ((5a+3b)-7)((5a+3b)+7) = (5a+3b)^2 - 49 = 25a^2 + 30ab + 9b^2 - 49$

問題7 (6): $(9a-7+4b)(9a-2+4b)$

$(9a-7+4b)(9a-2+4b) = (9a+4b-7)(9a+4b-2) = ((9a+4b) - 7)((9a+4b) - 2) = (9a+4b)^2 -9(9a+4b)+14 = (81a^2+72ab+16b^2)-81a-36b+14$

3. 最終的な答え

問題4:

(1) $x^2 + 14xy + 49y^2$

(2) $16x^2 - 8xy + y^2$

(3) $25x^2 + 20xy + 4y^2$

(4) $0.09a^2 - 0.12ab + 0.04b^2$

(5) $4a^2 - 20ab + 25b^2$

(6) $9x^2 + 12xy + 4y^2$

問題5:

(1) $x^2 - 121$

(2) $64 - a^2$

(3) $x^2 - 25y^2$

(4) $81a^2 - 4b^2$

(5) $a^2 - 0.01b^2$

(6) $36y^2 - 49x^2$

問題6:

(1) $x^2 - 10$

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

(3) $2x^2 - 18x + 41$

(4) $16a + 4$

(5) $23x^2 + 20x - 51$

(6) $32$

問題7:

(1) $x^2 + 9y^2 + 25 - 6xy + 10x - 30y$

(2) $a^2 + 4ab + 4b^2 - a - 2b - 20$

(3) $9x^2 + 24xy + 16y^2 + 9x + 12y - 54$

(4) $9x^2 -24xy + 16y^2 + 6x - 8y - 48$

(5) $25a^2 + 30ab + 9b^2 - 49$

(6) $81a^2+72ab+16b^2-81a-36b+14$