Whakaoti i te 0=30x^2+11x-30

$$0 = 30 x ^ { 2 } + 11 x - 30$$

Whakaoti mō x

$$x = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

$$x=\frac{5}{6}\approx 0.833333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$30x^{2}+11x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 30x^{2}+ax+bx-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=30\left(-30\right)=-900$$

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 -900.

$$-1,900 -2,450 -3,300 -4,225 -5,180 -6,150 -9,100 -10,90 -12,75 -15,60 -18,50 -20,45 -25,36 -30,30$$

Tātaihia te tapeke mō ia takirua.

$$-1+900=899 -2+450=448 -3+300=297 -4+225=221 -5+180=175 -6+150=144 -9+100=91 -10+90=80 -12+75=63 -15+60=45 -18+50=32 -20+45=25 -25+36=11 -30+30=0$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=-25 b=36$$

Tuhia anō te 30x^{2}+11x-30 hei \left(30x^{2}-25x\right)+\left(36x-30\right).

$$\left(30x^{2}-25x\right)+\left(36x-30\right)$$

Tauwehea te 5x i te tuatahi me te 6 i te rōpū tuarua.

$$5x\left(6x-5\right)+6\left(6x-5\right)$$

Whakatauwehea atu te kīanga pātahi 6x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x-5\right)\left(5x+6\right)$$

Hei kimi otinga whārite, me whakaoti te 6x-5=0 me te 5x+6=0.

$$x=\frac{5}{6} x=-\frac{6}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$30x^{2}+11x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 30 mō a, 11 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 30\left(-30\right)}}{2\times 30}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 30\left(-30\right)}}{2\times 30}$$

Whakareatia -4 ki te 30.

$$x=\frac{-11±\sqrt{121-120\left(-30\right)}}{2\times 30}$$

Whakareatia -120 ki te -30.

$$x=\frac{-11±\sqrt{121+3600}}{2\times 30}$$

Tāpiri 121 ki te 3600.

$$x=\frac{-11±\sqrt{3721}}{2\times 30}$$

Tuhia te pūtakerua o te 3721.

$$x=\frac{-11±61}{2\times 30}$$

Whakareatia 2 ki te 30.

$$x=\frac{-11±61}{60}$$

Nā, me whakaoti te whārite x=\frac{-11±61}{60} ina he tāpiri te ±. Tāpiri -11 ki te 61.

$$x=\frac{50}{60}$$

Whakahekea te hautanga \frac{50}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{5}{6}$$

Nā, me whakaoti te whārite x=\frac{-11±61}{60} ina he tango te ±. Tango 61 mai i -11.

$$x=-\frac{72}{60}$$

Whakahekea te hautanga \frac{-72}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$x=-\frac{6}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{6} x=-\frac{6}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$30x^{2}+11x-30=0$$

Me tāpiri te 30 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$30x^{2}+11x=30$$

Whakawehea ngā taha e rua ki te 30.

$$\frac{30x^{2}+11x}{30}=\frac{30}{30}$$

Mā te whakawehe ki te 30 ka wetekia te whakareanga ki te 30.

$$x^{2}+\frac{11}{30}x=\frac{30}{30}$$

Whakawehe 30 ki te 30.

$$x^{2}+\frac{11}{30}x=1$$

Whakawehea te \frac{11}{30}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{60}. Nā, tāpiria te pūrua o te \frac{11}{60} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{11}{30}x+\left(\frac{11}{60}\right)^{2}=1+\left(\frac{11}{60}\right)^{2}$$

Pūruatia \frac{11}{60} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{11}{30}x+\frac{121}{3600}=1+\frac{121}{3600}$$

Tāpiri 1 ki te \frac{121}{3600}.

$$x^{2}+\frac{11}{30}x+\frac{121}{3600}=\frac{3721}{3600}$$

Tauwehea x^{2}+\frac{11}{30}x+\frac{121}{3600}. 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}.

$$\left(x+\frac{11}{60}\right)^{2}=\frac{3721}{3600}$$

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

$$\sqrt{\left(x+\frac{11}{60}\right)^{2}}=\sqrt{\frac{3721}{3600}}$$

Whakarūnātia.

$$x+\frac{11}{60}=\frac{61}{60} x+\frac{11}{60}=-\frac{61}{60}$$

Me tango \frac{11}{60} mai i ngā taha e rua o te whārite.

$$x=\frac{5}{6} x=-\frac{6}{5}$$