32032中３・多項式の因数分解・計算問題・因数分解の公式2-2

2023年3月25日 2023年7月8日

banno

次の式を展開しなさい。

(1) $4 x^{2} + 8 x + 4$

(2) $9 x^{2} - 24 x + 16$

(3) $25 a^{2} + 60 a + 36$

(4) $100 x^{2} - 140 x + 49$

(5) $144 a^{2} - 72 a + 9$

解答・解説

公式２ $x^{2} + 2 a x + a^{2} = (x + a)^{2}$

公式３ $x^{2} - 2 a x + a^{2} = (x - a)^{2}$

(1) $\begin{array}{r} 4 x^{2} + 8 x + 4 \end{array}$

$\begin{array}{rcl} = & (2 x)^{2} + 2 \times 2 \times 2 x + 2^{2} \\ = & (2 x + 2)^{2} \end{array}$

答 $(2 x + 2)^{2}$

(2) $\begin{array}{r} 9 x^{2} - 24 x + 16 \end{array}$

$\begin{array}{rcl} = & (3 x)^{2} - 2 \times 4 \times 3 x + 4^{2} \\ = & (3 x - 4)^{2} \end{array}$

答 $(3 x - 4)^{2}$

(3) $\begin{array}{r} 25 a^{2} + 60 a + 36 \end{array}$

$\begin{array}{rcl} = & (5 a)^{2} + 2 \times 6 \times 5 a + 6^{2} \\ = & (5 a + 6)^{2} \end{array}$

答 $(5 a + 6)^{2}$

(4) $\begin{array}{r} 100 x^{2} - 140 x + 49 \end{array}$

$\begin{array}{rcl} = & (10 x)^{2} - 2 \times 7 \times 10 x + 7^{2} \\ = & (10 x - 7)^{2} \end{array}$

答 $(10 x - 7)^{2}$

(5) $\begin{array}{r} 144 a^{2} - 72 a + 9 \end{array}$

$\begin{array}{rcl} = & (12 a)^{2} - 2 \times 3 \times 12 a + 3^{2} \\ = & (12 a - 3)^{2} \end{array}$

答 $(12 a - 3)^{2}$