Tentukan himpunan penyelesaian persamaan \((x\:-\: 1)^{2x^2 \:-\: 3x \:-\: 2} = (3\:-\: x)^{2x^2 \:-\: 3x \:-\: 2}\)
Langkah 1: \(f(x) = g(x)\)
\(x\:-\: 1 = 3\:-\: x\)
\(x + x = 3 + 1\)
\(2x = 4\)
\(x = \dfrac{4}{2} = 2\)
Langkah 2: \(f(x) = -g(x)\), syarat \(h(x)\) genap
\(x\:-\: 1 = -(3\:-\: x)\)
\(x\:-\: 1 = -3+x\:\:\:\:\:\color{red}\text{tidak ada solusi}\)
Langkah 3: \(h(x) = 0\), dengan \(f(x), g(x) \neq 0\)
\(2x^2\:-\: 3x \:-\: 2 = 0\)
\((2x + 1)(x \:-\: 2) = 0\)
\(2x + 1 = 0\)
\(2x = -1\)
\(x_1 = -\frac{1}{2}\:\:\:\:\:\:\color{blue}\text{memenuhi}\)
\(f(-\frac{1}{2}) = -\frac{1}{2}- 1 \neq 0\)
\(g(-\frac{1}{2}) = 3 – (-\frac{1}{2}) \neq 0\)
\(x \:-\:2 = 0\)
\(x_2 = 2\:\:\:\:\:\:\color{blue}\text{memenuhi}\)
\(f(2) = 2 \:-\: 1 \neq 0\)
\(g(2) = 3 \:-\: 2 \neq 0\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -\frac{1}{2}, \:2\rbrace}\)
Tentukan himpunan penyelesaian persamaan \((x\:-\: 1)^{x^2 + 5x + 6} = (2x + 6)^{x^2 + 5x + 6}\)
Langkah 1: \(f(x) = g(x)\)
\(x\:-\: 1 = 2x + 6\)
\(x\:-\: 2x = 6 + 1\)
\(-x = 7\)
\(x = -7\:\:\:\:\:\color{blue} \text{memenuhi}\)
Langkah 2: \(f(x) = -g(x)\), syarat \(h(x)\) genap
\(x\:-\:1 = -(2x + 6)\)
\(x\:-\:1 = -2x – 6\)
\(x + 2x = -6 + 1\)
\(3x = -5\)
\(x = -\dfrac{5}{3}\)
\(h(-\dfrac{5}{3}) = (-\dfrac{5}{3})^2 + 5(-\dfrac{5}{3}) + 6\)
\(h(-\dfrac{5}{3}) = \dfrac{25}{9} -\dfrac{25}{3} + 6\)
\(h(-\dfrac{5}{3}) = \dfrac{4}{9}\:\:\:\:\:\color{red} \text{tidak genap}\)
\(\xcancel{x = -\dfrac{5}{3}}\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
Langkah 3: \(h(x) = 0\), dengan \(f(x), g(x) \neq 0\)
\(x^2 + 5x + 6 = 0\)
\((x + 3)(x + 2)= 0\)
\(x + 3 = 0\)
\(x_1 = -3\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
\(f(-3) = -3 \:-\: 1 \neq 0\)
\(g(-3) = 2(-3) + 6 = 0\)
\(x + 2 = 0\)
\(x_2 = -2\:\:\:\:\:\color{blue}\text{memenuhi}\)
\(f(-2) = -2\:-\:1 \neq 0\)
\(g(-2) = 2(-2) + 6 \neq 0\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace -7, \: -2\rbrace}\)
Tentukan himpunan penyelesaian persamaan \((x^2 + 2x \:-\: 3)^{x\:-\: 1} = (x^2 + x + 9)^{x\:-\: 1}\)
Langkah 1: \(f(x) = g(x)\)
\(\cancel{x^2} + 2x – 3 = \cancel{x^2} + x + 9\)
\(2x\:-\:x = 9 + 3\)
\(x = 12\:\:\:\:\:\color{blue} \text{memenuhi}\)
Langkah 2: \(f(x) = -g(x)\), syarat \(h(x)\) genap
\(x^2 + 2x \:-\: 3 = -(x^2 + x + 9)\)
\(x^2 + x^2 + 2x + x – 3 + 9 =0\)
\(2x^2 + 3x + 6 = 0\)
\(\text{Cek nilai diskriminan:}\)
\(b^2 \:-\:4ac = 3^2 \:-\: 4(2)(5) < 0\)
\(\color{red}\text{tidak ada solusi karena diskriminan negatif}\)
Langkah 3: \(h(x) = 0\), dengan \(f(x), g(x) \neq 0\)
\(x\:-\: 1= 0\)
\(x = 1\:\:\:\:\:\color{red}\text{tidak memenuhi}\)
\(\color{red} f(1) = 1^2 + 2(1) \:-\: 3 = 0\)
\(\bbox[yellow, 5px]{\text{HP} = \lbrace 12\rbrace}\)
