ແກ້ f(x)=(x-alpha)(x-beta)

$$f ( x ) = ( x - \alpha ) ( x - \beta )$$

ແກ້ສຳລັບ f (complex solution)

$$\left\{\begin{matrix}f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&\left(\alpha =0\text{ or }\beta =0\right)\text{ and }x=0\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ໃຊ້ຄຸນສົມບັດການແຈກແຈງເພື່ອຄູນ x-\alpha ດ້ວຍ x-\beta .

$$fx=x^{2}-x\beta -\alpha x+\beta \alpha$$

ສົມຜົນຢູ່ໃນຮູບແບບມາດຕະຖານ.

$$xf=x^{2}-x\alpha -x\beta +\alpha \beta$$

ຫານທັງສອງຂ້າງດ້ວຍ x.

$$\frac{xf}{x}=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}$$

ການຫານດ້ວຍ x ຈະຍົກເລີກການຄູນດ້ວຍ x.

$$f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}$$

ແກ້ສຳລັບ f

$$\left\{\begin{matrix}f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}\text{, }&x\neq 0\\f\in \mathrm{R}\text{, }&\left(\alpha =0\text{ or }\beta =0\right)\text{ and }x=0\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ໃຊ້ຄຸນສົມບັດການແຈກແຈງເພື່ອຄູນ x-\alpha ດ້ວຍ x-\beta .

$$fx=x^{2}-x\beta -\alpha x+\beta \alpha$$

ສົມຜົນຢູ່ໃນຮູບແບບມາດຕະຖານ.

$$xf=x^{2}-x\alpha -x\beta +\alpha \beta$$

ຫານທັງສອງຂ້າງດ້ວຍ x.

$$\frac{xf}{x}=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}$$

ການຫານດ້ວຍ x ຈະຍົກເລີກການຄູນດ້ວຍ x.

$$f=\frac{\left(x-\alpha \right)\left(x-\beta \right)}{x}$$

ແກ້ສຳລັບ x (complex solution)

$$x=\frac{-\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}$$

$$x=\frac{\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}$$

ແກ້ສຳລັບ x

$$x=\frac{-\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}$$

$$x=\frac{\sqrt{f^{2}+2f\alpha +2f\beta +\alpha ^{2}-2\alpha \beta +\beta ^{2}}+f+\alpha +\beta }{2}\text{, }\left(\alpha >0\text{ and }\beta <0\right)\text{ or }\left(\beta >0\text{ and }\alpha <0\right)\text{ or }f\geq 2\sqrt{\alpha \beta }-\alpha -\beta \text{ or }f\leq -2\sqrt{\alpha \beta }-\alpha -\beta \text{ or }\left(\beta \geq 0\text{ and }\alpha \leq 0\right)\text{ or }\left(\alpha \geq 0\text{ and }\beta \leq 0\right)$$