Whakaoti i te 2w^2+w-1275=0

$$2 w ^ { 2 } + w - 1275 = 0$$

Whakaoti mō w

$$w = -\frac{51}{2} = -25\frac{1}{2} = -25.5$$

$$w=25$$

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 2w^{2}+aw+bw-1275. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=2\left(-1275\right)=-2550$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -2550.

$$-1,2550 -2,1275 -3,850 -5,510 -6,425 -10,255 -15,170 -17,150 -25,102 -30,85 -34,75 -50,51$$

Tātaihia te tapeke mō ia takirua.

$$-1+2550=2549 -2+1275=1273 -3+850=847 -5+510=505 -6+425=419 -10+255=245 -15+170=155 -17+150=133 -25+102=77 -30+85=55 -34+75=41 -50+51=1$$

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

$$a=-50 b=51$$

Tuhia anō te 2w^{2}+w-1275 hei \left(2w^{2}-50w\right)+\left(51w-1275\right).

$$\left(2w^{2}-50w\right)+\left(51w-1275\right)$$

Tauwehea te 2w i te tuatahi me te 51 i te rōpū tuarua.

$$2w\left(w-25\right)+51\left(w-25\right)$$

Whakatauwehea atu te kīanga pātahi w-25 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(w-25\right)\left(2w+51\right)$$

Hei kimi otinga whārite, me whakaoti te w-25=0 me te 2w+51=0.

$$w=25 w=-\frac{51}{2}$$

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.

$$2w^{2}+w-1275=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 1 mō b, me -1275 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-1±\sqrt{1^{2}-4\times 2\left(-1275\right)}}{2\times 2}$$

Pūrua 1.

$$w=\frac{-1±\sqrt{1-4\times 2\left(-1275\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$w=\frac{-1±\sqrt{1-8\left(-1275\right)}}{2\times 2}$$

Whakareatia -8 ki te -1275.

$$w=\frac{-1±\sqrt{1+10200}}{2\times 2}$$

Tāpiri 1 ki te 10200.

$$w=\frac{-1±\sqrt{10201}}{2\times 2}$$

Tuhia te pūtakerua o te 10201.

$$w=\frac{-1±101}{2\times 2}$$

Whakareatia 2 ki te 2.

$$w=\frac{-1±101}{4}$$

Nā, me whakaoti te whārite w=\frac{-1±101}{4} ina he tāpiri te ±. Tāpiri -1 ki te 101.

$$w=\frac{100}{4}$$

Whakawehe 100 ki te 4.

$$w=25$$

Nā, me whakaoti te whārite w=\frac{-1±101}{4} ina he tango te ±. Tango 101 mai i -1.

$$w=-\frac{102}{4}$$

Whakahekea te hautanga \frac{-102}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$w=-\frac{51}{2}$$

Kua oti te whārite te whakatau.

$$w=25 w=-\frac{51}{2}$$

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.

$$2w^{2}+w-1275=0$$

Me tāpiri 1275 ki ngā taha e rua o te whārite.

$$2w^{2}+w-1275-\left(-1275\right)=-\left(-1275\right)$$

Mā te tango i te -1275 i a ia ake anō ka toe ko te 0.

$$2w^{2}+w=-\left(-1275\right)$$

Tango -1275 mai i 0.

$$2w^{2}+w=1275$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2w^{2}+w}{2}=\frac{1275}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$w^{2}+\frac{1}{2}w=\frac{1275}{2}$$

Whakawehea te \frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{4}. Nā, tāpiria te pūrua o te \frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}+\frac{1}{2}w+\left(\frac{1}{4}\right)^{2}=\frac{1275}{2}+\left(\frac{1}{4}\right)^{2}$$

Pūruatia \frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$w^{2}+\frac{1}{2}w+\frac{1}{16}=\frac{1275}{2}+\frac{1}{16}$$

Tāpiri \frac{1275}{2} ki te \frac{1}{16} 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.

$$w^{2}+\frac{1}{2}w+\frac{1}{16}=\frac{10201}{16}$$

Tauwehea w^{2}+\frac{1}{2}w+\frac{1}{16}. 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}.

$$\left(w+\frac{1}{4}\right)^{2}=\frac{10201}{16}$$

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

$$\sqrt{\left(w+\frac{1}{4}\right)^{2}}=\sqrt{\frac{10201}{16}}$$

Whakarūnātia.

$$w+\frac{1}{4}=\frac{101}{4} w+\frac{1}{4}=-\frac{101}{4}$$

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$w=25 w=-\frac{51}{2}$$