Whakaoti i te 2x^2+9x+7=3

$$2 x ^ { 2 } + 9 x + 7 = 3$$

Whakaoti mō x

$$x=-4$$

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

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3 mai i ngā taha e rua.

$$2x^{2}+9x+7-3=0$$

Tangohia te 3 i te 7, ka 4.

$$2x^{2}+9x+4=0$$

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.

$$a+b=9 ab=2\times 4=8$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 8.

$$1,8 2,4$$

Tātaihia te tapeke mō ia takirua.

$$1+8=9 2+4=6$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=1 b=8$$

Tuhia anō te 2x^{2}+9x+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.

$$2x^{2}+9x+7=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$2x^{2}+9x+7-3=3-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$2x^{2}+9x+7-3=0$$

Tango 3 mai i 7.

$$2x^{2}+9x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 9 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 2\times 4}}{2\times 2}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 2\times 4}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-9±\sqrt{81-8\times 4}}{2\times 2}$$

Whakareatia -8 ki te 4.

$$x=\frac{-9±\sqrt{81-32}}{2\times 2}$$

Tāpiri 81 ki te -32.

$$x=\frac{-9±\sqrt{49}}{2\times 2}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-9±7}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-9±7}{4}$$

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

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

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{-9±7}{4} ina he tango te ±. Tango 7 mai i -9.

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

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.

$$2x^{2}+9x+7=3$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$2x^{2}+9x+7-7=3-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$2x^{2}+9x=3-7$$

Tango 7 mai i 3.

$$2x^{2}+9x=-4$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+9x}{2}=-\frac{4}{2}$$

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

$$x^{2}+\frac{9}{2}x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x^{2}+\frac{9}{2}x=-2$$

Whakawehea te \frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{4}. Nā, tāpiria te pūrua o te \frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=-2+\left(\frac{9}{4}\right)^{2}$$

Pūruatia \frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=-2+\frac{81}{16}$$

Tāpiri -2 ki te \frac{81}{16}.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{49}{16}$$

Tauwehea x^{2}+\frac{9}{2}x+\frac{81}{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}.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{49}{16}$$

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

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Whakarūnātia.

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

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

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