Whakaoti i te c^2-10c-125=0

$${ c }^{ 2 } -10c-125 = 0$$

Whakaoti mō c

$$c=5\sqrt{6}+5\approx 17.247448714$$

$$c=5-5\sqrt{6}\approx -7.247448714$$

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.

$$c^{2}-10c-125=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me -125 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-125\right)}}{2}$$

Pūrua -10.

$$c=\frac{-\left(-10\right)±\sqrt{100-4\left(-125\right)}}{2}$$

Whakareatia -4 ki te -125.

$$c=\frac{-\left(-10\right)±\sqrt{100+500}}{2}$$

Tāpiri 100 ki te 500.

$$c=\frac{-\left(-10\right)±\sqrt{600}}{2}$$

Tuhia te pūtakerua o te 600.

$$c=\frac{-\left(-10\right)±10\sqrt{6}}{2}$$

Ko te tauaro o -10 ko 10.

$$c=\frac{10±10\sqrt{6}}{2}$$

Nā, me whakaoti te whārite c=\frac{10±10\sqrt{6}}{2} ina he tāpiri te ±. Tāpiri 10 ki te 10\sqrt{6}.

$$c=\frac{10\sqrt{6}+10}{2}$$

Whakawehe 10+10\sqrt{6} ki te 2.

$$c=5\sqrt{6}+5$$

Nā, me whakaoti te whārite c=\frac{10±10\sqrt{6}}{2} ina he tango te ±. Tango 10\sqrt{6} mai i 10.

$$c=\frac{10-10\sqrt{6}}{2}$$

Whakawehe 10-10\sqrt{6} ki te 2.

$$c=5-5\sqrt{6}$$

Kua oti te whārite te whakatau.

$$c=5\sqrt{6}+5 c=5-5\sqrt{6}$$

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.

$$c^{2}-10c-125=0$$

Me tāpiri 125 ki ngā taha e rua o te whārite.

$$c^{2}-10c-125-\left(-125\right)=-\left(-125\right)$$

Mā te tango i te -125 i a ia ake anō ka toe ko te 0.

$$c^{2}-10c=-\left(-125\right)$$

Tango -125 mai i 0.

$$c^{2}-10c=125$$

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

$$c^{2}-10c+\left(-5\right)^{2}=125+\left(-5\right)^{2}$$

Pūrua -5.

$$c^{2}-10c+25=125+25$$

Tāpiri 125 ki te 25.

$$c^{2}-10c+25=150$$

Tauwehea c^{2}-10c+25. 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}.

$$\left(c-5\right)^{2}=150$$

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

$$\sqrt{\left(c-5\right)^{2}}=\sqrt{150}$$

Whakarūnātia.

$$c-5=5\sqrt{6} c-5=-5\sqrt{6}$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$c=5\sqrt{6}+5 c=5-5\sqrt{6}$$