Persamaan kuadrat $a{x}^2+bx+c=$ memiliki akar-akar $x_1$ dan $x_2$, maka:

• $x_1+x_2=-{\displaystyle \frac{b}{a}}$
• $x_1.x_2={\displaystyle \frac{c}{a}}$
• $x_1-x_2=-{\displaystyle \frac{\sqrt{D}}{a}}$, untuk $x_1>x_2$

Contoh

Diketahui persamaan kuadrat $x^2+3x-5=0$

Tentukan nilai dari:

(A)   ${\displaystyle \frac{1}{x_1}}+{\displaystyle \frac{1}{x_2}}$

(B)   ${\displaystyle \frac{x_1}{x_2}}+{\displaystyle \frac{x_2}{x_1}}$

$x^2+3x-5=0$

(A)  ${\displaystyle \frac{1}{x_1}}+{\displaystyle \frac{1}{x_2}}$

$$\begin{array}{rl}& \frac{1}{x_1}+\frac{1}{x_2}\\ \\ & \frac{x_1+x_2}{x_1.x_2}\\ \\ & \frac{-3}{-5}\\ \\ & \frac{3}{5}\end{array}$$

(B)   ${\displaystyle \frac{x_1}{x_2}}+{\displaystyle \frac{x_2}{x_1}}$

$$\begin{array}{rl}& \frac{x_1}{x_2}+\frac{x_2}{x_1}\\ \\ & \frac{x_1^2+x_2^2}{x_1.x_2}\\ \\ & \frac{(x_1+x_2{)}^2-2.x_1.x_2}{x_1.x_2}\\ \\ & \frac{(-3{)}^2-2.(-5)}{-5}\\ \\ & \frac{9+10}{-5}\\ \\ & -\frac{19}{5}\end{array}$$