HUBUNGAN AKAR-AKAR
Persamaan kuadrat \(ax^2 + bx + c \ =\) memiliki akar-akar \(x_1\) dan \(x_2\), maka:
- \(x_1 + x_2 = - \dfrac{b}{a}\)
- \(x_1 \:.\: x_2 = \dfrac{c}{a}\)
- \(x_1 - x_2 = - \dfrac{\sqrt{D}}{a}\), untuk \(x_1 > x_2\)
Contoh
Diketahui persamaan kuadrat \(x^2 + 3x - 5 = 0\)
Tentukan nilai dari:
(A) \(\dfrac{1}{x_1} + \dfrac{1}{x_2} \)
(B) \(\dfrac{x_1}{x_2} + \dfrac{x_2}{x_1} \)
\(x^2 + 3x - 5 = 0\)
\begin{equation*} \begin{split} & x_1 + x_2 = - \frac ba \\\\ & x_1 + x_2 = - \frac 31 \\\\ & \bbox[5px, border: 2px solid blue] {x_1 + x_2 = -3} \end{split} \end{equation*}
\begin{equation*} \begin{split} & x_1 \:.\: x_2 = \frac ca \\\\ & x_1 \:.\: x_2 = \frac {-5}{1} \\\\ & \bbox[5px, border: 2px solid blue] {x_1 \:.\: x_2 = -5} \end{split} \end{equation*}
(A) \(\dfrac{1}{x_1} + \dfrac{1}{x_2} \)
\begin{equation*} \begin{split} & \frac{1}{x_1} + \frac{1}{x_2} \\\\ & \frac{x_1 + x_2}{x_1 \:.\: x_2} \\\\ & \frac{-3}{-5} \\\\ & \bbox[5px, border: 2px solid magenta] {\frac 35} \end{split} \end{equation*}
(B) \(\dfrac{x_1}{x_2} + \dfrac{x_2}{x_1} \)
\begin{equation*} \begin{split} & \frac{x_1}{x_2} + \frac{x_2}{x_1} \\\\ & \frac{x_1^2 + x_2^2}{x_1 \:.\: x_2} \\\\ & \frac{(x_1 + x_2)^2 - 2 \:.\: x_1 \:.\: x_2}{x_1 \:.\: x_2} \\\\ & \frac{(-3)^2 - 2 \:.\: (-5)}{-5} \\\\ & \frac{9 + 10}{-5} \\\\ & \bbox[5px, border: 2px solid magenta] {- \frac {19}{5}} \end{split} \end{equation*}
SOAL LATIHAN