\(5^{x^2 \:-\: 3} = 7^{x^2 \:-\: 3}\)
\(x^2 \:-\: 3 = 0\:\:\:\:\:\color{blue}\text{faktorkan}\)
\(x^2\:-\: (\sqrt{3})^2 = 0\:\:\:\:\:\color{blue} a^2 \:-\: b^2 = (a + b)(a\:-\: b)\)
\((x + \sqrt{3})(x \:-\: \sqrt{3}) = 0\)
\(x_1 = -\sqrt{3}\)
\(x_2 = \sqrt{3}\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -\sqrt{3}, \sqrt{3}\rbrace}\)
Tentukan himpunan penyelesaian persamaan \((0,125)^{4-3x – x^2} = 3^{3x^2 + 9x -12}\)
\((0,125)^{4\:-\: 3x \:-\: x^2} = 3^{3x^2 + 9x \:-\: 12}\)
\(\Big(\dfrac{1}{8}\Big)^{4\:-\: 3x\:-\: x^2} = 3^{3x^2 + 9x \:-\: 12}\)
\(\Big(\dfrac{1}{2^3}\Big)^{4\:-\: 3x \:-\: x^2} = 3^{3x^2 + 9x\:-\: 12}\)
\(2^{-3(4\:-\: 3x \:-\: x^2)} = 3^{3x^2 + 9x \:-\: 12}\)
\(2^{3x^2 + 9x \:-\: 12} = 3^{3x^2 + 9x \:-\: 12}\)
\(3x^2 + 9x \:-\: 12 = 0\)
\(\color{blue}\text{bagi kedua ruas dengan 3}\)
\(x^2 + 3x \:-\: 4 = 0\)
\((x + 4)(x \:-\: 1) = 0\)
\(x + 4 = 0\)
\(x_1 = -4\)
\(x – 1 = 0\)
\(x_2 = 1\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -4, 1\rbrace}\)
Tentukan himpunan penyelesaian persamaan \(2^{x\:-\: 3}\cdot 5^{2x + 7} = 2^{-1\:-\: x^2}\cdot 5^{x^2 + 3x + 5}\)
\(2^{x\:-\: 3}\cdot 5^{2x + 7} = 2^{-1\:-\: x^2}\cdot 5^{x^2 + 3x + 5}\)
\(\dfrac{5^{2x + 7}}{5^{x^2 + 3x + 5}} = \dfrac{2^{-1\:-\: x^2}}{2^{x\:-\: 3}}\)
\(5^{2x + 7\:-\: (x^2 + 3x + 5)} = 2^{-1-x^2\:-\: (x \:-\: 3)}\)
\(5^{2x + 7\:-\: x^2\:-\: 3x\:-\: 5} = 2^{-1\:-\: x^2\:-\: x + 3}\)
\(5^{-x^2\:-\: x + 2} = 2^{-x^2 \:-\: x + 2}\)
\(-x^2 \:-\: x + 2 = 0\)
\(\color{blue}\text{kalikan kedua ruas dengan } -1\)
\(x^2 + x\:-\: 2 = 0\)
\((x + 2)(x \:-\: 1) = 0\)
\(x_1 = -2\)
\(x_2 = 1\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -2, 1\rbrace}\)
