Whakaoti i te 3x^2+16x-35=0

$$3 x ^ { 2 } + 16 x - 35 = 0$$

Whakaoti mō x

$$x=-7$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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-35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=16 ab=3\left(-35\right)=-105$$

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 -105.

$$-1,105 -3,35 -5,21 -7,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=-5 b=21$$

Tuhia anō te 3x^{2}+16x-35 hei \left(3x^{2}-5x\right)+\left(21x-35\right).

$$\left(3x^{2}-5x\right)+\left(21x-35\right)$$

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

$$x\left(3x-5\right)+7\left(3x-5\right)$$

Whakatauwehea atu te kīanga pātahi 3x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-5\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-5=0 me te x+7=0.

$$x=\frac{5}{3} x=-7$$

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.

$$3x^{2}+16x-35=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 16 mō b, me -35 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 3\left(-35\right)}}{2\times 3}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 3\left(-35\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-16±\sqrt{256-12\left(-35\right)}}{2\times 3}$$

Whakareatia -12 ki te -35.

$$x=\frac{-16±\sqrt{256+420}}{2\times 3}$$

Tāpiri 256 ki te 420.

$$x=\frac{-16±\sqrt{676}}{2\times 3}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-16±26}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-16±26}{6}$$

Nā, me whakaoti te whārite x=\frac{-16±26}{6} ina he tāpiri te ±. Tāpiri -16 ki te 26.

$$x=\frac{10}{6}$$

Whakahekea te hautanga \frac{10}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{5}{3}$$

Nā, me whakaoti te whārite x=\frac{-16±26}{6} ina he tango te ±. Tango 26 mai i -16.

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

Whakawehe -42 ki te 6.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{3} x=-7$$

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.

$$3x^{2}+16x-35=0$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$3x^{2}+16x-35-\left(-35\right)=-\left(-35\right)$$

Mā te tango i te -35 i a ia ake anō ka toe ko te 0.

$$3x^{2}+16x=-\left(-35\right)$$

Tango -35 mai i 0.

$$3x^{2}+16x=35$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+16x}{3}=\frac{35}{3}$$

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

$$x^{2}+\frac{16}{3}x=\frac{35}{3}$$

Whakawehea te \frac{16}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{8}{3}. Nā, tāpiria te pūrua o te \frac{8}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=\frac{35}{3}+\left(\frac{8}{3}\right)^{2}$$

Pūruatia \frac{8}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{35}{3}+\frac{64}{9}$$

Tāpiri \frac{35}{3} ki te \frac{64}{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.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{169}{9}$$

Tauwehea x^{2}+\frac{16}{3}x+\frac{64}{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}.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{169}{9}$$

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

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{169}{9}}$$

Whakarūnātia.

$$x+\frac{8}{3}=\frac{13}{3} x+\frac{8}{3}=-\frac{13}{3}$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{5}{3} x=-7$$