Whakaoti i te 5x^2-x-2184=0

$$5 { x }^{ 2 } -x-2184=0$$

Whakaoti mō x

$$x = -\frac{104}{5} = -20\frac{4}{5} = -20.8$$

$$x=21$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5x^{2}+ax+bx-2184. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=5\left(-2184\right)=-10920$$

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 -10920.

$$1,-10920 2,-5460 3,-3640 4,-2730 5,-2184 6,-1820 7,-1560 8,-1365 10,-1092 12,-910 13,-840 14,-780 15,-728 20,-546 21,-520 24,-455 26,-420 28,-390 30,-364 35,-312 39,-280 40,-273 42,-260 52,-210 56,-195 60,-182 65,-168 70,-156 78,-140 84,-130 91,-120 104,-105$$

Tātaihia te tapeke mō ia takirua.

$$1-10920=-10919 2-5460=-5458 3-3640=-3637 4-2730=-2726 5-2184=-2179 6-1820=-1814 7-1560=-1553 8-1365=-1357 10-1092=-1082 12-910=-898 13-840=-827 14-780=-766 15-728=-713 20-546=-526 21-520=-499 24-455=-431 26-420=-394 28-390=-362 30-364=-334 35-312=-277 39-280=-241 40-273=-233 42-260=-218 52-210=-158 56-195=-139 60-182=-122 65-168=-103 70-156=-86 78-140=-62 84-130=-46 91-120=-29 104-105=-1$$

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

$$a=-105 b=104$$

Tuhia anō te 5x^{2}-x-2184 hei \left(5x^{2}-105x\right)+\left(104x-2184\right).

$$\left(5x^{2}-105x\right)+\left(104x-2184\right)$$

Tauwehea te 5x i te tuatahi me te 104 i te rōpū tuarua.

$$5x\left(x-21\right)+104\left(x-21\right)$$

Whakatauwehea atu te kīanga pātahi x-21 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-21\right)\left(5x+104\right)$$

Hei kimi otinga whārite, me whakaoti te x-21=0 me te 5x+104=0.

$$x=21 x=-\frac{104}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$5x^{2}-x-2184=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -1 mō b, me -2184 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 5\left(-2184\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-1\right)±\sqrt{1-20\left(-2184\right)}}{2\times 5}$$

Whakareatia -20 ki te -2184.

$$x=\frac{-\left(-1\right)±\sqrt{1+43680}}{2\times 5}$$

Tāpiri 1 ki te 43680.

$$x=\frac{-\left(-1\right)±\sqrt{43681}}{2\times 5}$$

Tuhia te pūtakerua o te 43681.

$$x=\frac{-\left(-1\right)±209}{2\times 5}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±209}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{1±209}{10}$$

Nā, me whakaoti te whārite x=\frac{1±209}{10} ina he tāpiri te ±. Tāpiri 1 ki te 209.

$$x=\frac{210}{10}$$

Whakawehe 210 ki te 10.

$$x=21$$

Nā, me whakaoti te whārite x=\frac{1±209}{10} ina he tango te ±. Tango 209 mai i 1.

$$x=-\frac{208}{10}$$

Whakahekea te hautanga \frac{-208}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{104}{5}$$

Kua oti te whārite te whakatau.

$$x=21 x=-\frac{104}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$5x^{2}-x-2184=0$$

Me tāpiri 2184 ki ngā taha e rua o te whārite.

$$5x^{2}-x-2184-\left(-2184\right)=-\left(-2184\right)$$

Mā te tango i te -2184 i a ia ake anō ka toe ko te 0.

$$5x^{2}-x=-\left(-2184\right)$$

Tango -2184 mai i 0.

$$5x^{2}-x=2184$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-x}{5}=\frac{2184}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{1}{5}x=\frac{2184}{5}$$

Whakawehea te -\frac{1}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{10}. Nā, tāpiria te pūrua o te -\frac{1}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{5}x+\left(-\frac{1}{10}\right)^{2}=\frac{2184}{5}+\left(-\frac{1}{10}\right)^{2}$$

Pūruatia -\frac{1}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{2184}{5}+\frac{1}{100}$$

Tāpiri \frac{2184}{5} ki te \frac{1}{100} 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.

$$x^{2}-\frac{1}{5}x+\frac{1}{100}=\frac{43681}{100}$$

Tauwehea x^{2}-\frac{1}{5}x+\frac{1}{100}. 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}.

$$\left(x-\frac{1}{10}\right)^{2}=\frac{43681}{100}$$

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

$$\sqrt{\left(x-\frac{1}{10}\right)^{2}}=\sqrt{\frac{43681}{100}}$$

Whakarūnātia.

$$x-\frac{1}{10}=\frac{209}{10} x-\frac{1}{10}=-\frac{209}{10}$$

Me tāpiri \frac{1}{10} ki ngā taha e rua o te whārite.

$$x=21 x=-\frac{104}{5}$$