Whakaoti i te 15x^2+11x+2=0

$$15 x ^ { 2 } + 11 x + 2 = 0$$

Whakaoti mō x

$$x=-\frac{2}{5}=-0.4$$

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

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 15x^{2}+ax+bx+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=15\times 2=30$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 30.

$$1,30 2,15 3,10 5,6$$

Tātaihia te tapeke mō ia takirua.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

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

$$a=5 b=6$$

Tuhia anō te 15x^{2}+11x+2 hei \left(15x^{2}+5x\right)+\left(6x+2\right).

$$\left(15x^{2}+5x\right)+\left(6x+2\right)$$

Tauwehea te 5x i te tuatahi me te 2 i te rōpū tuarua.

$$5x\left(3x+1\right)+2\left(3x+1\right)$$

Whakatauwehea atu te kīanga pātahi 3x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+1\right)\left(5x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+1=0 me te 5x+2=0.

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

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.

$$15x^{2}+11x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, 11 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 15\times 2}}{2\times 15}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 15\times 2}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-11±\sqrt{121-60\times 2}}{2\times 15}$$

Whakareatia -60 ki te 2.

$$x=\frac{-11±\sqrt{121-120}}{2\times 15}$$

Tāpiri 121 ki te -120.

$$x=\frac{-11±\sqrt{1}}{2\times 15}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-11±1}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{-11±1}{30}$$

Nā, me whakaoti te whārite x=\frac{-11±1}{30} ina he tāpiri te ±. Tāpiri -11 ki te 1.

$$x=-\frac{10}{30}$$

Whakahekea te hautanga \frac{-10}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

Nā, me whakaoti te whārite x=\frac{-11±1}{30} ina he tango te ±. Tango 1 mai i -11.

$$x=-\frac{12}{30}$$

Whakahekea te hautanga \frac{-12}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{2}{5}$$

Kua oti te whārite te whakatau.

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

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.

$$15x^{2}+11x+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$15x^{2}+11x+2-2=-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$15x^{2}+11x=-2$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}+11x}{15}=-\frac{2}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}+\frac{11}{15}x=-\frac{2}{15}$$

Whakawehea te \frac{11}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{30}. Nā, tāpiria te pūrua o te \frac{11}{30} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{11}{15}x+\left(\frac{11}{30}\right)^{2}=-\frac{2}{15}+\left(\frac{11}{30}\right)^{2}$$

Pūruatia \frac{11}{30} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{11}{15}x+\frac{121}{900}=-\frac{2}{15}+\frac{121}{900}$$

Tāpiri -\frac{2}{15} ki te \frac{121}{900} 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.

$$x^{2}+\frac{11}{15}x+\frac{121}{900}=\frac{1}{900}$$

Tauwehea x^{2}+\frac{11}{15}x+\frac{121}{900}. 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}.

$$\left(x+\frac{11}{30}\right)^{2}=\frac{1}{900}$$

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

$$\sqrt{\left(x+\frac{11}{30}\right)^{2}}=\sqrt{\frac{1}{900}}$$

Whakarūnātia.

$$x+\frac{11}{30}=\frac{1}{30} x+\frac{11}{30}=-\frac{1}{30}$$

Me tango \frac{11}{30} mai i ngā taha e rua o te whārite.

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