Whakaoti i te 3x^2-7x+4=0

$$3 x ^ { 2 } - 7 x + 4 = 0$$

Whakaoti mō x

$$x=1$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

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 3x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=3\times 4=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

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

$$a=-4 b=-3$$

Tuhia anō te 3x^{2}-7x+4 hei \left(3x^{2}-4x\right)+\left(-3x+4\right).

$$\left(3x^{2}-4x\right)+\left(-3x+4\right)$$

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

$$x\left(3x-4\right)-\left(3x-4\right)$$

Whakatauwehea atu te kīanga pātahi 3x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-4\right)\left(x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-4=0 me te x-1=0.

$$x=\frac{4}{3} x=1$$

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.

$$3x^{2}-7x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -7 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\times 4}}{2\times 3}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\times 4}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\times 4}}{2\times 3}$$

Whakareatia -12 ki te 4.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2\times 3}$$

Tāpiri 49 ki te -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2\times 3}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-\left(-7\right)±1}{2\times 3}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±1}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{7±1}{6}$$

Nā, me whakaoti te whārite x=\frac{7±1}{6} ina he tāpiri te ±. Tāpiri 7 ki te 1.

$$x=\frac{8}{6}$$

Whakahekea te hautanga \frac{8}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{4}{3}$$

Nā, me whakaoti te whārite x=\frac{7±1}{6} ina he tango te ±. Tango 1 mai i 7.

$$x=\frac{6}{6}$$

Whakawehe 6 ki te 6.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=\frac{4}{3} x=1$$

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.

$$3x^{2}-7x+4=0$$

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

$$3x^{2}-7x+4-4=-4$$

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

$$3x^{2}-7x=-4$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-7x}{3}=-\frac{4}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{7}{3}x=-\frac{4}{3}$$

Whakawehea te -\frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{6}. Nā, tāpiria te pūrua o te -\frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=-\frac{4}{3}+\left(-\frac{7}{6}\right)^{2}$$

Pūruatia -\frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=-\frac{4}{3}+\frac{49}{36}$$

Tāpiri -\frac{4}{3} ki te \frac{49}{36} 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.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{1}{36}$$

Tauwehea te x^{2}-\frac{7}{3}x+\frac{49}{36}. 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}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{1}{36}$$

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

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Whakarūnātia.

$$x-\frac{7}{6}=\frac{1}{6} x-\frac{7}{6}=-\frac{1}{6}$$

Me tāpiri \frac{7}{6} ki ngā taha e rua o te whārite.

$$x=\frac{4}{3} x=1$$