FAKTORISASI
Bentuk \(ax^2 + bx = 0\)
\begin{equation*} \begin{split} & ax^2 + bx = 0 \\\\ & x \:.\: (ax + b) = 0 \\\\ & x = 0 \text{ atau } ax + b = 0 \end{split} \end{equation*}
Bentuk \(x^2 − a^2 = 0\)
\begin{equation*} \begin{split} & x^2 – a^2 = 0 \\\\ & (x + a) \:.\: (x – a) = 0 \\\\ & x + a = 0 \text{ atau } x – a = 0 \\\\ & x = -a \text{ atau } x = a \end{split} \end{equation*}
Bentuk \(ax^2 + bx + c = 0\)
Lakukan cara trial and error (lihat contoh soal)
Contoh 01
\begin{equation*} \begin{split} & x^2 – 5x = 0 \\\\ & x \:.\: (x – 5) = 0 \\\\ & x = 0 \text{ atau } x = 5 \end{split} \end{equation*}
Contoh 02
\begin{equation*} \begin{split} & x^2 – 9 = 0 \\\\ & x^2 – 3^2 = 0 \\\\ & (x + 3)(x – 3) = 0 \\\\ & x + 3 = 0 \text{ atau } x – 3 = 0 \\\\ & x = -3 \text{ atau } x = 3 \end{split} \end{equation*}
Contoh 03
\begin{equation*} \begin{split} & x^2 + 3x + 2 = 0 \\\\ & (x + 2)(x + 1) = 0 \\\\ & x + 2 = 0 \text{ atau } x + 1 & = 0 \\\\ & x = -2 \text{ atau } x = -1 \end{split} \end{equation*}
SOAL LATIHAN
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Tentukan akar-akar persamaan di bawah ini!
\(5x^2 + 15x = 0\)
\(5x^2 + 15x = 0\)
Pembahasan:
\begin{equation*} \begin{split} & 5x^2 + 15x = 0 \\\\ & 5x \:.\: (x + 3) = 0 \\\\ & 5x = 0 \text{ atau } x + 3 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 0 \text{ atau } x = -3} \end{split} \end{equation*}
\(4x^2 – 25 = 0\)
\(4x^2 – 25 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & 4x^2 – 25 = 0 \\\\ & (2x)^2 – 5^2 = 0 \\\\ & (2x + 5)(2x – 5) = 0 \\\\ & 2x + 5 = 0 \text{ atau } 2x – 5 = 0 \\\\ & 2x = -5 \text{ atau } 2x = 5 \\\\ & \bbox[5px, border: 2px solid magenta] {x = -2\frac{1}{2} \text{ atau } x = 2\frac{1}{2}} \end{split} \end{equation*}
\(x^2 + 5x + 6 = 0\)
\(x^2 – x – 6 = 0\)
\(x^2 – 7x + 6 = 0\)
\(x^2 + 5x – 6 = 0\)
\(x^2 + 5x + 6 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & x^2 + 5x + 6 = 0 \\\\ & (x + 3)(x + 2) = 0 \\\\ & x + 3 = 0 \text{ atau } x + 2 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = -3 \text{ atau } x = -2} \end{split} \end{equation*}
\(x^2 – 3x – 10 = 0\)
\(x^2 – 7x + 10 = 0\)
\(x^2 – 11x + 10 = 0\)
\(x^2 + 9x – 10 = 0\)
\(x^2 – 3x – 10 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & x^2 – 3x – 10 = 0 \\\\ & (x – 5)(x + 2) = 0 \\\\ & x – 5 = 0 \text{ atau } x + 2 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 5 \text{ atau } x = -2} \end{split} \end{equation*}
\(x^2 – 9x + 14 = 0\)
\(x^2 + 13x – 14 = 0\)
\(x^2 – 9x + 14 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & x^2 – 9x + 14 = 0 \\\\ & (x – 2)(x – 7) = 0 \\\\ & x – 2 = 0 \text{ atau } x – 7 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 2 \text{ atau } x = 7} \end{split} \end{equation*}
\(2x^2 – 7x + 6 = 0\)
\(2x^2 – 11x – 6 = 0\)
\(2x^2 – 7x + 6 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & 2x^2 – 7x + 6 = 0 \\\\ & (2x – 3)(x – 2) = 0 \\\\ & 2x – 3 = 0 \text{ atau } x – 2 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = \frac{3}{2} \text{ atau } x = 2} \end{split} \end{equation*}
\(-3x^2 + 11x – 6 = 0\)
\(-3x^2 + 11x – 6 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & -3x^2 + 11x – 6 = 0 \quad {\color {blue} \times (-1)} \\\\ & 3x^2 – 11x + 6 = 0 \\\\ & (3x – 2)(x – 3) = 0 \\\\ & 3x – 2 = 0 \text{ atau } x – 3 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = \frac{2}{3} \text{ atau } x = 3} \end{split} \end{equation*}
\(6x^2 + 13x – 28 = 0\)
\(6x^2 + 13x – 28 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & 6x^2 + 13x – 28 = 0 \\\\ & (6x – 7)(x + 4) = 0 \\\\ & 6x – 7 = 0 \text{ atau } x + 4 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = \frac{7}{6} \text{ atau } x = -4} \end{split} \end{equation*}
\(15x^2 + 16x – 119 = 0\)
\(15x^2 + 16x – 119 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & 15x^2 + 16x – 119 = 0 \\\\ & (5x + 17)(3x – 7) = 0 \\\\ & 5x + 17 = 0 \text{ atau } 3x – 7 & = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = -\frac{17}{5} \text{ atau } x = \frac{7}{3}} \end{split} \end{equation*}
\(-12x^2 – x + 20 = 0\)
\(-12x^2 – x + 20 = 0\)
Pembahasan:
\begin{equation*} \begin{split} & -12x^2 – x + 20 = 0 \quad {\color {blue} \times (-1)}\\\\ & 12x^2 + x – 20 = 0 \\\\ & (4x – 5)(3x + 4) = 0 \\\\ & 4x – 5 = 0 \text{ atau } 3x + 4 = 0 \\\\ & \bbox[5px, border: 2px solid magenta] {x = \frac{5}{4} \text{ atau } x = -\frac{4}{3}} \end{split} \end{equation*}
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