Točno
8. prosinca 2013. 16:15 (10 godine, 7 mjeseci)
Sakrij rješenje
Sakrij rješenje
Upozorenje: Ovaj zadatak još niste riješili!
Kliknite ovdje kako biste prikazali rješenje.
Kliknite ovdje kako biste prikazali rješenje.
supstituiramo
, primjetimo da je tada i
, te zbog uvjeta zadatka znamo da
nije rjesenje. Mnozenjem s
, i grupiranjem za kvadratnu po
slijedi:
![y^2 ( t^3 ) + y ( 2t^2 ) + ( t^2 - t + 1 ) = 0](/media/m/6/4/8/648a3927b6fbad644c22acf4f1476a17.png)
diskriminantna ove jednadbe mora biti nenegativna buduci da trazimo realne
, dakle mora biti
![4t^4 - 4t^5 + 4t^4 - 4t^3 \geq 0](/media/m/3/d/e/3ded4de9c469529cd74fe284b295a5d1.png)
to jest
![-4t^3( t - 1 )^2 \geq 0](/media/m/c/2/5/c25dd5cfbe28bbc8712b5f4cda57c7a9.png)
zbog
ovo je moguce jedino kad je
. tada je
, tj
, te
, tj
ili
.
sve u svemu, imamo
rjesenja,
.
![t = 4x^2 + 4x + 1](/media/m/6/5/a/65a55d0a671e535e2d09904461b95f7c.png)
![t = ( 2x + 1 )^2 \geq 0](/media/m/7/8/6/786059e2127a9ab7d6f7f0b4f57d134d.png)
![t = 0](/media/m/7/9/5/795c8aacba1e23e70fd158777264afc1.png)
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![y^2 ( t^3 ) + y ( 2t^2 ) + ( t^2 - t + 1 ) = 0](/media/m/6/4/8/648a3927b6fbad644c22acf4f1476a17.png)
diskriminantna ove jednadbe mora biti nenegativna buduci da trazimo realne
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
![4t^4 - 4t^5 + 4t^4 - 4t^3 \geq 0](/media/m/3/d/e/3ded4de9c469529cd74fe284b295a5d1.png)
to jest
![-4t^3( t - 1 )^2 \geq 0](/media/m/c/2/5/c25dd5cfbe28bbc8712b5f4cda57c7a9.png)
zbog
![t \geq 0](/media/m/e/e/6/ee6869e3c517492b2f328274f5f98c4d.png)
![t=1](/media/m/8/8/b/88bc171f31d0f68acba1f66054f9b027.png)
![y^2 + 2y + 1 = 0](/media/m/f/f/0/ff0e212934e6ec0e9bcabe7f137beca3.png)
![y = -1](/media/m/2/5/2/252f6f0884d172676b92581543251da9.png)
![4x^2 + 4x + 1 = 1](/media/m/9/4/3/943859ad900f8d58407c1dea07cb7e5a.png)
![x = 0](/media/m/8/2/6/8268205131b210915acd475f4122137c.png)
![x = -1](/media/m/9/9/e/99e75c7cadfc6ae8ff3cdb7f9eb54bb2.png)
sve u svemu, imamo
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![(x,y) = (0,-1), (-1,-1)](/media/m/8/1/0/810d122b07e511806a1c6170052fb504.png)