Whakaoti i te 60x^2-44x+4=0

$$60 x ^ { 2 } - 44 x + 4 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{61}+11}{30}\approx 0.627008323$$

$$x=\frac{11-\sqrt{61}}{30}\approx 0.106325011$$

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.

$$60x^{2}-44x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 60 mō a, -44 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 60\times 4}}{2\times 60}$$

Pūrua -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\times 60\times 4}}{2\times 60}$$

Whakareatia -4 ki te 60.

$$x=\frac{-\left(-44\right)±\sqrt{1936-240\times 4}}{2\times 60}$$

Whakareatia -240 ki te 4.

$$x=\frac{-\left(-44\right)±\sqrt{1936-960}}{2\times 60}$$

Tāpiri 1936 ki te -960.

$$x=\frac{-\left(-44\right)±\sqrt{976}}{2\times 60}$$

Tuhia te pūtakerua o te 976.

$$x=\frac{-\left(-44\right)±4\sqrt{61}}{2\times 60}$$

Ko te tauaro o -44 ko 44.

$$x=\frac{44±4\sqrt{61}}{2\times 60}$$

Whakareatia 2 ki te 60.

$$x=\frac{44±4\sqrt{61}}{120}$$

Nā, me whakaoti te whārite x=\frac{44±4\sqrt{61}}{120} ina he tāpiri te ±. Tāpiri 44 ki te 4\sqrt{61}.

$$x=\frac{4\sqrt{61}+44}{120}$$

Whakawehe 44+4\sqrt{61} ki te 120.

$$x=\frac{\sqrt{61}+11}{30}$$

Nā, me whakaoti te whārite x=\frac{44±4\sqrt{61}}{120} ina he tango te ±. Tango 4\sqrt{61} mai i 44.

$$x=\frac{44-4\sqrt{61}}{120}$$

Whakawehe 44-4\sqrt{61} ki te 120.

$$x=\frac{11-\sqrt{61}}{30}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{61}+11}{30} x=\frac{11-\sqrt{61}}{30}$$

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.

$$60x^{2}-44x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$60x^{2}-44x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$60x^{2}-44x=-4$$

Whakawehea ngā taha e rua ki te 60.

$$\frac{60x^{2}-44x}{60}=-\frac{4}{60}$$

Mā te whakawehe ki te 60 ka wetekia te whakareanga ki te 60.

$$x^{2}+\left(-\frac{44}{60}\right)x=-\frac{4}{60}$$

Whakahekea te hautanga \frac{-44}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{11}{15}x=-\frac{4}{60}$$

Whakahekea te hautanga \frac{-4}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{11}{15}x=-\frac{1}{15}$$

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.

$$x^{2}-\frac{11}{15}x+\left(-\frac{11}{30}\right)^{2}=-\frac{1}{15}+\left(-\frac{11}{30}\right)^{2}$$

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

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

Tāpiri -\frac{1}{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.

$$x^{2}-\frac{11}{15}x+\frac{121}{900}=\frac{61}{900}$$

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}.

$$\left(x-\frac{11}{30}\right)^{2}=\frac{61}{900}$$

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

$$\sqrt{\left(x-\frac{11}{30}\right)^{2}}=\sqrt{\frac{61}{900}}$$

Whakarūnātia.

$$x-\frac{11}{30}=\frac{\sqrt{61}}{30} x-\frac{11}{30}=-\frac{\sqrt{61}}{30}$$

Me tāpiri \frac{11}{30} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{61}+11}{30} x=\frac{11-\sqrt{61}}{30}$$