Whakaoti i te 2x^2-9x+4=0

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

Whakaoti mō x

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

$$x=4$$

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.

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

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro 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.

$$a=-8 b=-1$$

Tuhia anō te 2x^{2}-9x+4 hei \left(2x^{2}-8x\right)+\left(-x+4\right).

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

Tauwehea te 2x i te tuatahi me te -1 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

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

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}.

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

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\times 4}}{2\times 2}$$

Whakareatia -4 ki te 2.

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

Whakareatia -8 ki te 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 2}$$

Tāpiri 81 ki te -32.

$$x=\frac{-\left(-9\right)±\sqrt{49}}{2\times 2}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-9\right)±7}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±7}{2\times 2}$$

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.

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

Whakawehe 16 ki te 4.

$$x=4$$

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

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

Kua oti te whārite te whakatau.

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

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.

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

Me tango 4 mai i ngā taha e rua o te whārite.

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

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

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

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 tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

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