Whakaoti i te 3x^2=22x-7

$$3 x ^ { 2 } = 22 x - 7$$

Whakaoti mō x

$$x=\frac{1}{3}\approx 0.333333333$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 22x mai i ngā taha e rua.

$$3x^{2}-22x=-7$$

Me tāpiri te 7 ki ngā taha e rua.

$$3x^{2}-22x+7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-22 ab=3\times 7=21$$

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 21.

$$-1,-21 -3,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-21=-22 -3-7=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -22.

$$a=-21 b=-1$$

Tuhia anō te 3x^{2}-22x+7 hei \left(3x^{2}-21x\right)+\left(-x+7\right).

$$\left(3x^{2}-21x\right)+\left(-x+7\right)$$

Tauwehea te 3x i te tuatahi me te -1 i te rōpū tuarua.

$$3x\left(x-7\right)-\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(3x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te 3x-1=0.

$$x=7 x=\frac{1}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 22x mai i ngā taha e rua.

$$3x^{2}-22x=-7$$

Me tāpiri te 7 ki ngā taha e rua.

$$3x^{2}-22x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -22 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 3\times 7}}{2\times 3}$$

Pūrua -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 3\times 7}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-22\right)±\sqrt{484-12\times 7}}{2\times 3}$$

Whakareatia -12 ki te 7.

$$x=\frac{-\left(-22\right)±\sqrt{484-84}}{2\times 3}$$

Tāpiri 484 ki te -84.

$$x=\frac{-\left(-22\right)±\sqrt{400}}{2\times 3}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-22\right)±20}{2\times 3}$$

Ko te tauaro o -22 ko 22.

$$x=\frac{22±20}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{22±20}{6}$$

Nā, me whakaoti te whārite x=\frac{22±20}{6} ina he tāpiri te ±. Tāpiri 22 ki te 20.

$$x=\frac{42}{6}$$

Whakawehe 42 ki te 6.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{22±20}{6} ina he tango te ±. Tango 20 mai i 22.

$$x=\frac{2}{6}$$

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

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

Kua oti te whārite te whakatau.

$$x=7 x=\frac{1}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 22x mai i ngā taha e rua.

$$3x^{2}-22x=-7$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-22x}{3}=-\frac{7}{3}$$

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

$$x^{2}-\frac{22}{3}x=-\frac{7}{3}$$

Whakawehea te -\frac{22}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{3}. Nā, tāpiria te pūrua o te -\frac{11}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{22}{3}x+\left(-\frac{11}{3}\right)^{2}=-\frac{7}{3}+\left(-\frac{11}{3}\right)^{2}$$

Pūruatia -\frac{11}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{7}{3}+\frac{121}{9}$$

Tāpiri -\frac{7}{3} ki te \frac{121}{9} 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.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=\frac{100}{9}$$

Tauwehea x^{2}-\frac{22}{3}x+\frac{121}{9}. 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}.

$$\left(x-\frac{11}{3}\right)^{2}=\frac{100}{9}$$

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

$$\sqrt{\left(x-\frac{11}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Whakarūnātia.

$$x-\frac{11}{3}=\frac{10}{3} x-\frac{11}{3}=-\frac{10}{3}$$

Me tāpiri \frac{11}{3} ki ngā taha e rua o te whārite.

$$x=7 x=\frac{1}{3}$$