Tentukan himpunan penyelesaian persamaan \(2^{3x^2 + 2x + 5} = 2^{x^2 \:-\: 2x + 11}\)
\(2^{3x^2 + 2x + 5} = 2^{x^2\:-\: 2x + 11}\)
\(3x^2 + 2x + 5 = x^2 \:-\: 2x + 11\)
\(3x^2 \:-\: x^2 + 2x + 2x + 5 \:-\: 11 = 0\)
\(2x^2 + 4x \:-\: 6 = 0\)
\(\color{blue}\text{bagi kedua ruas dengan 2}\)
\(x^2 + 2x \:-\: 3 = 0\:\:\:\:\:\color{blue}\text{faktorkan}\)
\((x + 3)(x\:-\: 1)=0\)
\(x + 3 = 0 \rightarrow x = -3\)
\(x – 1 = 0 \rightarrow x = 1\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -3, 1 \rbrace}\)
Tentukan himpunan penyelesaian persamaan \(27^{x^2 \:-\: 5} = (\sqrt{3})^{2x^2 + 14x \:-\: 22}\)
\(27^{x^2\:-\: 5} = (\sqrt{3})^{2x^2 + 14x \:-\: 22}\:\:\:\:\:\color{blue}\sqrt{3} = 3^{\frac{1}{2}}\)
\(3^{3(x^2 \:-\: 5)} = 3^{\frac{1}{2}(2x^2 + 14x \:-\: 22)}\)
\(3(x^2\:-\: 5) = \frac{1}{2}(2x^2 + 14x\:-\: 22)\)
\(3x^2 \:-\: 15 = x^2 + 7x \:-\: 11\)
\(3x^2 \:-\: x^2 \:-\: 7x\:-\: 15 + 11 = 0\)
\(2x^2 \:-\: 7x \:-\: 4 = 0\:\:\:\:\:\color{blue}\text{faktorkan}\)
\((2x + 1)(x \:-\: 4)=0\)
\(2x + 1 = 0\)
\(2x = -1\)
\(x_1 = -\dfrac{1}{2}\)
\(x – 4 = 0\)
\(x_2 = 4\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -\dfrac{1}{2}, 4\rbrace}\)
Tentukan himpunan penyelesaian persamaan \(\sqrt[3]{\dfrac{1}{729^x}} = \dfrac{(3^{4x})^3}{81^{x + 5}}\)
\(\sqrt[3]{\dfrac{1}{729^x}} = \dfrac{(3^{4x})^3}{81^{x + 5}}\)
\(\sqrt[3]{\dfrac{1}{3^{6x}}} = \dfrac{(3^{4x})^3}{3^{4(x + 5)}}\)
\(\Big(\dfrac{1}{3^{6x}}\Big)^{\frac{1}{3}} = \dfrac{3^{12x}}{3^{4x + 20}}\)
\(\dfrac{1^{\frac{1}{3}}}{3^{\frac{1}{3}(6x)}}= 3^{12x\:-\: (4x + 20)}\)
\(\dfrac{1}{3^{2x}} = 3^{12x\:-\: (4x + 20)}\)
\(3^{-2x} = 3^{8x – 20}\)
\(-2x = 8x \:-\: 20\)
\(-2x \:-\: 8x = – 20\)
\(-10x = -20\)
\(x = \dfrac{20}{10}\)
\(x = 2\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace 2\rbrace}\)
