Whakaoti i te 2x^2+5x-817=0

$$2 { x }^{ 2 } +5x-817 = 0$$

Whakaoti mō x

$$x = -\frac{43}{2} = -21\frac{1}{2} = -21.5$$

$$x=19$$

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 2x^{2}+ax+bx-817. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=2\left(-817\right)=-1634$$

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 -1634.

$$-1,1634 -2,817 -19,86 -38,43$$

Tātaihia te tapeke mō ia takirua.

$$-1+1634=1633 -2+817=815 -19+86=67 -38+43=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-38 b=43$$

Tuhia anō te 2x^{2}+5x-817 hei \left(2x^{2}-38x\right)+\left(43x-817\right).

$$\left(2x^{2}-38x\right)+\left(43x-817\right)$$

Tauwehea te 2x i te tuatahi me te 43 i te rōpū tuarua.

$$2x\left(x-19\right)+43\left(x-19\right)$$

Whakatauwehea atu te kīanga pātahi x-19 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-19\right)\left(2x+43\right)$$

Hei kimi otinga whārite, me whakaoti te x-19=0 me te 2x+43=0.

$$x=19 x=-\frac{43}{2}$$

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.

$$2x^{2}+5x-817=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 5 mō b, me -817 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-817\right)}}{2\times 2}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-817\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-5±\sqrt{25-8\left(-817\right)}}{2\times 2}$$

Whakareatia -8 ki te -817.

$$x=\frac{-5±\sqrt{25+6536}}{2\times 2}$$

Tāpiri 25 ki te 6536.

$$x=\frac{-5±\sqrt{6561}}{2\times 2}$$

Tuhia te pūtakerua o te 6561.

$$x=\frac{-5±81}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-5±81}{4}$$

Nā, me whakaoti te whārite x=\frac{-5±81}{4} ina he tāpiri te ±. Tāpiri -5 ki te 81.

$$x=\frac{76}{4}$$

Whakawehe 76 ki te 4.

$$x=19$$

Nā, me whakaoti te whārite x=\frac{-5±81}{4} ina he tango te ±. Tango 81 mai i -5.

$$x=-\frac{86}{4}$$

Whakahekea te hautanga \frac{-86}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{43}{2}$$

Kua oti te whārite te whakatau.

$$x=19 x=-\frac{43}{2}$$

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.

$$2x^{2}+5x-817=0$$

Me tāpiri 817 ki ngā taha e rua o te whārite.

$$2x^{2}+5x-817-\left(-817\right)=-\left(-817\right)$$

Mā te tango i te -817 i a ia ake anō ka toe ko te 0.

$$2x^{2}+5x=-\left(-817\right)$$

Tango -817 mai i 0.

$$2x^{2}+5x=817$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+5x}{2}=\frac{817}{2}$$

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

$$x^{2}+\frac{5}{2}x=\frac{817}{2}$$

Whakawehea te \frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{4}. Nā, tāpiria te pūrua o te \frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{817}{2}+\left(\frac{5}{4}\right)^{2}$$

Pūruatia \frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{817}{2}+\frac{25}{16}$$

Tāpiri \frac{817}{2} ki te \frac{25}{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.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{6561}{16}$$

Tauwehea x^{2}+\frac{5}{2}x+\frac{25}{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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{6561}{16}$$

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

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{6561}{16}}$$

Whakarūnātia.

$$x+\frac{5}{4}=\frac{81}{4} x+\frac{5}{4}=-\frac{81}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$x=19 x=-\frac{43}{2}$$