Whakaoti i te 42x^2-696x+3240=0

$$42 { x }^{ 2 } -696x+3240=0$$

Whakaoti mō x (complex solution)

$$x=\frac{58+4\sqrt{26}i}{7}\approx 8.285714286+2.913725436i$$

$$x=\frac{-4\sqrt{26}i+58}{7}\approx 8.285714286-2.913725436i$$

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.

$$42x^{2}-696x+3240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 42 mō a, -696 mō b, me 3240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-696\right)±\sqrt{\left(-696\right)^{2}-4\times 42\times 3240}}{2\times 42}$$

Pūrua -696.

$$x=\frac{-\left(-696\right)±\sqrt{484416-4\times 42\times 3240}}{2\times 42}$$

Whakareatia -4 ki te 42.

$$x=\frac{-\left(-696\right)±\sqrt{484416-168\times 3240}}{2\times 42}$$

Whakareatia -168 ki te 3240.

$$x=\frac{-\left(-696\right)±\sqrt{484416-544320}}{2\times 42}$$

Tāpiri 484416 ki te -544320.

$$x=\frac{-\left(-696\right)±\sqrt{-59904}}{2\times 42}$$

Tuhia te pūtakerua o te -59904.

$$x=\frac{-\left(-696\right)±48\sqrt{26}i}{2\times 42}$$

Ko te tauaro o -696 ko 696.

$$x=\frac{696±48\sqrt{26}i}{2\times 42}$$

Whakareatia 2 ki te 42.

$$x=\frac{696±48\sqrt{26}i}{84}$$

Nā, me whakaoti te whārite x=\frac{696±48\sqrt{26}i}{84} ina he tāpiri te ±. Tāpiri 696 ki te 48i\sqrt{26}.

$$x=\frac{696+48\sqrt{26}i}{84}$$

Whakawehe 696+48i\sqrt{26} ki te 84.

$$x=\frac{58+4\sqrt{26}i}{7}$$

Nā, me whakaoti te whārite x=\frac{696±48\sqrt{26}i}{84} ina he tango te ±. Tango 48i\sqrt{26} mai i 696.

$$x=\frac{-48\sqrt{26}i+696}{84}$$

Whakawehe 696-48i\sqrt{26} ki te 84.

$$x=\frac{-4\sqrt{26}i+58}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}$$

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.

$$42x^{2}-696x+3240=0$$

Me tango 3240 mai i ngā taha e rua o te whārite.

$$42x^{2}-696x+3240-3240=-3240$$

Mā te tango i te 3240 i a ia ake anō ka toe ko te 0.

$$42x^{2}-696x=-3240$$

Whakawehea ngā taha e rua ki te 42.

$$\frac{42x^{2}-696x}{42}=-\frac{3240}{42}$$

Mā te whakawehe ki te 42 ka wetekia te whakareanga ki te 42.

$$x^{2}+\left(-\frac{696}{42}\right)x=-\frac{3240}{42}$$

Whakahekea te hautanga \frac{-696}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}-\frac{116}{7}x=-\frac{3240}{42}$$

Whakahekea te hautanga \frac{-3240}{42} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x^{2}-\frac{116}{7}x=-\frac{540}{7}$$

Whakawehea te -\frac{116}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{58}{7}. Nā, tāpiria te pūrua o te -\frac{58}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{116}{7}x+\left(-\frac{58}{7}\right)^{2}=-\frac{540}{7}+\left(-\frac{58}{7}\right)^{2}$$

Pūruatia -\frac{58}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{540}{7}+\frac{3364}{49}$$

Tāpiri -\frac{540}{7} ki te \frac{3364}{49} 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.

$$x^{2}-\frac{116}{7}x+\frac{3364}{49}=-\frac{416}{49}$$

Tauwehea x^{2}-\frac{116}{7}x+\frac{3364}{49}. 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}.

$$\left(x-\frac{58}{7}\right)^{2}=-\frac{416}{49}$$

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

$$\sqrt{\left(x-\frac{58}{7}\right)^{2}}=\sqrt{-\frac{416}{49}}$$

Whakarūnātia.

$$x-\frac{58}{7}=\frac{4\sqrt{26}i}{7} x-\frac{58}{7}=-\frac{4\sqrt{26}i}{7}$$

Me tāpiri \frac{58}{7} ki ngā taha e rua o te whārite.

$$x=\frac{58+4\sqrt{26}i}{7} x=\frac{-4\sqrt{26}i+58}{7}$$