Whakaoti i te 5x^2-11x+4=0

$$5 x ^ { 2 } - 11 x + 4 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{41} + 11}{10} \approx 1.740312424$$

$$x=\frac{11-\sqrt{41}}{10}\approx 0.459687576$$

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.

$$5x^{2}-11x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -11 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 5\times 4}}{2\times 5}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 5\times 4}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-11\right)±\sqrt{121-20\times 4}}{2\times 5}$$

Whakareatia -20 ki te 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-80}}{2\times 5}$$

Tāpiri 121 ki te -80.

$$x=\frac{-\left(-11\right)±\sqrt{41}}{2\times 5}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{41}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{11±\sqrt{41}}{10}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{41}}{10} ina he tāpiri te ±. Tāpiri 11 ki te \sqrt{41}.

$$x=\frac{\sqrt{41}+11}{10}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{41}}{10} ina he tango te ±. Tango \sqrt{41} mai i 11.

$$x=\frac{11-\sqrt{41}}{10}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{41}+11}{10} x=\frac{11-\sqrt{41}}{10}$$

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.

$$5x^{2}-11x+4=0$$

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

$$5x^{2}-11x+4-4=-4$$

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

$$5x^{2}-11x=-4$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-11x}{5}=-\frac{4}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}-\frac{11}{5}x=-\frac{4}{5}$$

Whakawehea te -\frac{11}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{10}. Nā, tāpiria te pūrua o te -\frac{11}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{5}x+\left(-\frac{11}{10}\right)^{2}=-\frac{4}{5}+\left(-\frac{11}{10}\right)^{2}$$

Pūruatia -\frac{11}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{5}x+\frac{121}{100}=-\frac{4}{5}+\frac{121}{100}$$

Tāpiri -\frac{4}{5} ki te \frac{121}{100} 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.

$$x^{2}-\frac{11}{5}x+\frac{121}{100}=\frac{41}{100}$$

Tauwehea x^{2}-\frac{11}{5}x+\frac{121}{100}. 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}.

$$\left(x-\frac{11}{10}\right)^{2}=\frac{41}{100}$$

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

$$\sqrt{\left(x-\frac{11}{10}\right)^{2}}=\sqrt{\frac{41}{100}}$$

Whakarūnātia.

$$x-\frac{11}{10}=\frac{\sqrt{41}}{10} x-\frac{11}{10}=-\frac{\sqrt{41}}{10}$$

Me tāpiri \frac{11}{10} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{41}+11}{10} x=\frac{11-\sqrt{41}}{10}$$