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

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

Whakaoti mō x

$$x = -\frac{9}{7} = -1\frac{2}{7} \approx -1.285714286$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 9 mai i ngā taha e rua.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 7x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=7\left(-9\right)=-63$$

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 -63.

$$-1,63 -3,21 -7,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+63=62 -3+21=18 -7+9=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-7 b=9$$

Tuhia anō te 7x^{2}+2x-9 hei \left(7x^{2}-7x\right)+\left(9x-9\right).

$$\left(7x^{2}-7x\right)+\left(9x-9\right)$$

Tauwehea te 7x i te tuatahi me te 9 i te rōpū tuarua.

$$7x\left(x-1\right)+9\left(x-1\right)$$

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

$$\left(x-1\right)\left(7x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 7x+9=0.

$$x=1 x=-\frac{9}{7}$$

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.

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

Me tango 9 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 2 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 7\left(-9\right)}}{2\times 7}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 7\left(-9\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-2±\sqrt{4-28\left(-9\right)}}{2\times 7}$$

Whakareatia -28 ki te -9.

$$x=\frac{-2±\sqrt{4+252}}{2\times 7}$$

Tāpiri 4 ki te 252.

$$x=\frac{-2±\sqrt{256}}{2\times 7}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-2±16}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-2±16}{14}$$

Nā, me whakaoti te whārite x=\frac{-2±16}{14} ina he tāpiri te ±. Tāpiri -2 ki te 16.

$$x=\frac{14}{14}$$

Whakawehe 14 ki te 14.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-2±16}{14} ina he tango te ±. Tango 16 mai i -2.

$$x=-\frac{18}{14}$$

Whakahekea te hautanga \frac{-18}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

$$x=1 x=-\frac{9}{7}$$

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.

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

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+2x}{7}=\frac{9}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}+\frac{2}{7}x=\frac{9}{7}$$

Whakawehea te \frac{2}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{7}. Nā, tāpiria te pūrua o te \frac{1}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{7}x+\left(\frac{1}{7}\right)^{2}=\frac{9}{7}+\left(\frac{1}{7}\right)^{2}$$

Pūruatia \frac{1}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=\frac{9}{7}+\frac{1}{49}$$

Tāpiri \frac{9}{7} ki te \frac{1}{49} 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.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=\frac{64}{49}$$

Tauwehea x^{2}+\frac{2}{7}x+\frac{1}{49}. 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}.

$$\left(x+\frac{1}{7}\right)^{2}=\frac{64}{49}$$

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

$$\sqrt{\left(x+\frac{1}{7}\right)^{2}}=\sqrt{\frac{64}{49}}$$

Whakarūnātia.

$$x+\frac{1}{7}=\frac{8}{7} x+\frac{1}{7}=-\frac{8}{7}$$

Me tango \frac{1}{7} mai i ngā taha e rua o te whārite.

$$x=1 x=-\frac{9}{7}$$