Whakaoti i te 2019x^2=2020+[x]

$$2019 x ^ { 2 } = 2020 + [ x ]$$

Whakaoti mō x

$$x=-1$$

$$x = \frac{2020}{2019} = 1\frac{1}{2019} \approx 1.000495295$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 2020 mai i ngā taha e rua.

$$2019x^{2}-2020=x$$

Tangohia te x mai i ngā taha e rua.

$$2019x^{2}-2020-x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$2019x^{2}-x-2020=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2019x^{2}+ax+bx-2020. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2019\left(-2020\right)=-4078380$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -4078380.

$$1,-4078380 2,-2039190 3,-1359460 4,-1019595 5,-815676 6,-679730 10,-407838 12,-339865 15,-271892 20,-203919 30,-135946 60,-67973 101,-40380 202,-20190 303,-13460 404,-10095 505,-8076 606,-6730 673,-6060 1010,-4038 1212,-3365 1346,-3030 1515,-2692 2019,-2020$$

Tātaihia te tapeke mō ia takirua.

$$1-4078380=-4078379 2-2039190=-2039188 3-1359460=-1359457 4-1019595=-1019591 5-815676=-815671 6-679730=-679724 10-407838=-407828 12-339865=-339853 15-271892=-271877 20-203919=-203899 30-135946=-135916 60-67973=-67913 101-40380=-40279 202-20190=-19988 303-13460=-13157 404-10095=-9691 505-8076=-7571 606-6730=-6124 673-6060=-5387 1010-4038=-3028 1212-3365=-2153 1346-3030=-1684 1515-2692=-1177 2019-2020=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-2020 b=2019$$

Tuhia anō te 2019x^{2}-x-2020 hei \left(2019x^{2}-2020x\right)+\left(2019x-2020\right).

$$\left(2019x^{2}-2020x\right)+\left(2019x-2020\right)$$

Whakatauwehea atu x i te 2019x^{2}-2020x.

$$x\left(2019x-2020\right)+2019x-2020$$

Whakatauwehea atu te kīanga pātahi 2019x-2020 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2019x-2020\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 2019x-2020=0 me te x+1=0.

$$x=\frac{2020}{2019} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 2020 mai i ngā taha e rua.

$$2019x^{2}-2020=x$$

Tangohia te x mai i ngā taha e rua.

$$2019x^{2}-2020-x=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$2019x^{2}-x-2020=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2019 mō a, -1 mō b, me -2020 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2019\left(-2020\right)}}{2\times 2019}$$

Whakareatia -4 ki te 2019.

$$x=\frac{-\left(-1\right)±\sqrt{1-8076\left(-2020\right)}}{2\times 2019}$$

Whakareatia -8076 ki te -2020.

$$x=\frac{-\left(-1\right)±\sqrt{1+16313520}}{2\times 2019}$$

Tāpiri 1 ki te 16313520.

$$x=\frac{-\left(-1\right)±\sqrt{16313521}}{2\times 2019}$$

Tuhia te pūtakerua o te 16313521.

$$x=\frac{-\left(-1\right)±4039}{2\times 2019}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±4039}{2\times 2019}$$

Whakareatia 2 ki te 2019.

$$x=\frac{1±4039}{4038}$$

Nā, me whakaoti te whārite x=\frac{1±4039}{4038} ina he tāpiri te ±. Tāpiri 1 ki te 4039.

$$x=\frac{4040}{4038}$$

Whakahekea te hautanga \frac{4040}{4038} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{2020}{2019}$$

Nā, me whakaoti te whārite x=\frac{1±4039}{4038} ina he tango te ±. Tango 4039 mai i 1.

$$x=-\frac{4038}{4038}$$

Whakawehe -4038 ki te 4038.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{2020}{2019} x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te x mai i ngā taha e rua.

$$2019x^{2}-x=2020$$

Whakawehea ngā taha e rua ki te 2019.

$$\frac{2019x^{2}-x}{2019}=\frac{2020}{2019}$$

Mā te whakawehe ki te 2019 ka wetekia te whakareanga ki te 2019.

$$x^{2}-\frac{1}{2019}x=\frac{2020}{2019}$$

Whakawehea te -\frac{1}{2019}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4038}. Nā, tāpiria te pūrua o te -\frac{1}{4038} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2019}x+\left(-\frac{1}{4038}\right)^{2}=\frac{2020}{2019}+\left(-\frac{1}{4038}\right)^{2}$$

Pūruatia -\frac{1}{4038} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2019}x+\frac{1}{16305444}=\frac{2020}{2019}+\frac{1}{16305444}$$

Tāpiri \frac{2020}{2019} ki te \frac{1}{16305444} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{1}{2019}x+\frac{1}{16305444}=\frac{16313521}{16305444}$$

Tauwehea x^{2}-\frac{1}{2019}x+\frac{1}{16305444}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4038}\right)^{2}=\frac{16313521}{16305444}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{1}{4038}\right)^{2}}=\sqrt{\frac{16313521}{16305444}}$$

Whakarūnātia.

$$x-\frac{1}{4038}=\frac{4039}{4038} x-\frac{1}{4038}=-\frac{4039}{4038}$$

Me tāpiri \frac{1}{4038} ki ngā taha e rua o te whārite.

$$x=\frac{2020}{2019} x=-1$$