### Hubungan akar-akar

###### 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

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