Whakaoti i te 86t^2-76t+17=0

$$86 t ^ { 2 } - 76 t + 17 = 0$$

Whakaoti mō t

$$t=\frac{3\sqrt{2}i}{86}+\frac{19}{43}\approx 0.441860465+0.049333031i$$

$$t=-\frac{3\sqrt{2}i}{86}+\frac{19}{43}\approx 0.441860465-0.049333031i$$

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.

$$86t^{2}-76t+17=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 86 mō a, -76 mō b, me 17 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 86\times 17}}{2\times 86}$$

Pūrua -76.

$$t=\frac{-\left(-76\right)±\sqrt{5776-4\times 86\times 17}}{2\times 86}$$

Whakareatia -4 ki te 86.

$$t=\frac{-\left(-76\right)±\sqrt{5776-344\times 17}}{2\times 86}$$

Whakareatia -344 ki te 17.

$$t=\frac{-\left(-76\right)±\sqrt{5776-5848}}{2\times 86}$$

Tāpiri 5776 ki te -5848.

$$t=\frac{-\left(-76\right)±\sqrt{-72}}{2\times 86}$$

Tuhia te pūtakerua o te -72.

$$t=\frac{-\left(-76\right)±6\sqrt{2}i}{2\times 86}$$

Ko te tauaro o -76 ko 76.

$$t=\frac{76±6\sqrt{2}i}{2\times 86}$$

Whakareatia 2 ki te 86.

$$t=\frac{76±6\sqrt{2}i}{172}$$

Nā, me whakaoti te whārite t=\frac{76±6\sqrt{2}i}{172} ina he tāpiri te ±. Tāpiri 76 ki te 6i\sqrt{2}.

$$t=\frac{76+6\sqrt{2}i}{172}$$

Whakawehe 76+6i\sqrt{2} ki te 172.

$$t=\frac{3\sqrt{2}i}{86}+\frac{19}{43}$$

Nā, me whakaoti te whārite t=\frac{76±6\sqrt{2}i}{172} ina he tango te ±. Tango 6i\sqrt{2} mai i 76.

$$t=\frac{-6\sqrt{2}i+76}{172}$$

Whakawehe 76-6i\sqrt{2} ki te 172.

$$t=-\frac{3\sqrt{2}i}{86}+\frac{19}{43}$$

Kua oti te whārite te whakatau.

$$t=\frac{3\sqrt{2}i}{86}+\frac{19}{43} t=-\frac{3\sqrt{2}i}{86}+\frac{19}{43}$$

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.

$$86t^{2}-76t+17=0$$

Me tango 17 mai i ngā taha e rua o te whārite.

$$86t^{2}-76t+17-17=-17$$

Mā te tango i te 17 i a ia ake anō ka toe ko te 0.

$$86t^{2}-76t=-17$$

Whakawehea ngā taha e rua ki te 86.

$$\frac{86t^{2}-76t}{86}=-\frac{17}{86}$$

Mā te whakawehe ki te 86 ka wetekia te whakareanga ki te 86.

$$t^{2}+\left(-\frac{76}{86}\right)t=-\frac{17}{86}$$

Whakahekea te hautanga \frac{-76}{86} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$t^{2}-\frac{38}{43}t=-\frac{17}{86}$$

Whakawehea te -\frac{38}{43}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{43}. Nā, tāpiria te pūrua o te -\frac{19}{43} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{38}{43}t+\left(-\frac{19}{43}\right)^{2}=-\frac{17}{86}+\left(-\frac{19}{43}\right)^{2}$$

Pūruatia -\frac{19}{43} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{38}{43}t+\frac{361}{1849}=-\frac{17}{86}+\frac{361}{1849}$$

Tāpiri -\frac{17}{86} ki te \frac{361}{1849} 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.

$$t^{2}-\frac{38}{43}t+\frac{361}{1849}=-\frac{9}{3698}$$

Tauwehea t^{2}-\frac{38}{43}t+\frac{361}{1849}. 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}.

$$\left(t-\frac{19}{43}\right)^{2}=-\frac{9}{3698}$$

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

$$\sqrt{\left(t-\frac{19}{43}\right)^{2}}=\sqrt{-\frac{9}{3698}}$$

Whakarūnātia.

$$t-\frac{19}{43}=\frac{3\sqrt{2}i}{86} t-\frac{19}{43}=-\frac{3\sqrt{2}i}{86}$$

Me tāpiri \frac{19}{43} ki ngā taha e rua o te whārite.

$$t=\frac{3\sqrt{2}i}{86}+\frac{19}{43} t=-\frac{3\sqrt{2}i}{86}+\frac{19}{43}$$