Whakaoti i te 2x^2+7x-4=0

$$2 x ^ { 2 } + 7 x - 4 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=\frac{1}{2}=0.5$$

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-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=2\left(-4\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-1 b=8$$

Tuhia anō te 2x^{2}+7x-4 hei \left(2x^{2}-x\right)+\left(8x-4\right).

$$\left(2x^{2}-x\right)+\left(8x-4\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(2x-1\right)+4\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-1=0 me te x+4=0.

$$x=\frac{1}{2} x=-4$$

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.

$$2x^{2}+7x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 7 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\left(-4\right)}}{2\times 2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 2\left(-4\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-7±\sqrt{49-8\left(-4\right)}}{2\times 2}$$

Whakareatia -8 ki te -4.

$$x=\frac{-7±\sqrt{49+32}}{2\times 2}$$

Tāpiri 49 ki te 32.

$$x=\frac{-7±\sqrt{81}}{2\times 2}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-7±9}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-7±9}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{4} ina he tāpiri te ±. Tāpiri -7 ki te 9.

$$x=\frac{2}{4}$$

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

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{4} ina he tango te ±. Tango 9 mai i -7.

$$x=-\frac{16}{4}$$

Whakawehe -16 ki te 4.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=-4$$

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.

$$2x^{2}+7x-4=0$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x^{2}+7x-4-\left(-4\right)=-\left(-4\right)$$

Mā te tango i te -4 i a ia ake anō ka toe ko te 0.

$$2x^{2}+7x=-\left(-4\right)$$

Tango -4 mai i 0.

$$2x^{2}+7x=4$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+7x}{2}=\frac{4}{2}$$

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

$$x^{2}+\frac{7}{2}x=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$x^{2}+\frac{7}{2}x=2$$

Whakawehea te \frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{4}. Nā, tāpiria te pūrua o te \frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=2+\left(\frac{7}{4}\right)^{2}$$

Pūruatia \frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=2+\frac{49}{16}$$

Tāpiri 2 ki te \frac{49}{16}.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{81}{16}$$

Tauwehea te x^{2}+\frac{7}{2}x+\frac{49}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$x+\frac{7}{4}=\frac{9}{4} x+\frac{7}{4}=-\frac{9}{4}$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{2} x=-4$$