Whakaoti i te 0=3x^2+2x-5

$$0 = 3 x ^ { 2 } + 2 x - 5$$

Whakaoti mō x

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

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+2x-5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=3\left(-5\right)=-15$$

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 -15.

$$-1,15 -3,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+15=14 -3+5=2$$

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

$$a=-3 b=5$$

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

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

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

$$3x\left(x-1\right)+5\left(x-1\right)$$

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

$$\left(x-1\right)\left(3x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 3x+5=0.

$$x=1 x=-\frac{5}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+2x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 2 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 3\left(-5\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-2±\sqrt{4-12\left(-5\right)}}{2\times 3}$$

Whakareatia -12 ki te -5.

$$x=\frac{-2±\sqrt{4+60}}{2\times 3}$$

Tāpiri 4 ki te 60.

$$x=\frac{-2±\sqrt{64}}{2\times 3}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-2±8}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-2±8}{6}$$

Nā, me whakaoti te whārite x=\frac{-2±8}{6} ina he tāpiri te ±. Tāpiri -2 ki te 8.

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

Whakawehe 6 ki te 6.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-2±8}{6} ina he tango te ±. Tango 8 mai i -2.

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

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

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

Kua oti te whārite te whakatau.

$$x=1 x=-\frac{5}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3x^{2}+2x-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x^{2}+2x=5$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+2x}{3}=\frac{5}{3}$$

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

$$x^{2}+\frac{2}{3}x=\frac{5}{3}$$

Whakawehea te \frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{3}. Nā, tāpiria te pūrua o te \frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\frac{5}{3}+\left(\frac{1}{3}\right)^{2}$$

Pūruatia \frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{5}{3}+\frac{1}{9}$$

Tāpiri \frac{5}{3} ki te \frac{1}{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.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{16}{9}$$

Tauwehea x^{2}+\frac{2}{3}x+\frac{1}{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}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{16}{9}$$

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

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Whakarūnātia.

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

Me tango \frac{1}{3} mai i ngā taha e rua o te whārite.

$$x=1 x=-\frac{5}{3}$$