Whakaoti i te x^2-9x+10=0

$$x ^ { 2 } - 9 x + 10 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{41} + 9}{2} \approx 7.701562119$$

$$x = \frac{9 - \sqrt{41}}{2} \approx 1.298437881$$

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.

$$x^{2}-9x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -9 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 10}}{2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 10}}{2}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-9\right)±\sqrt{81-40}}{2}$$

Tāpiri 81 ki te -40.

$$x=\frac{-\left(-9\right)±\sqrt{41}}{2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±\sqrt{41}}{2}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{41}}{2} ina he tāpiri te ±. Tāpiri 9 ki te \sqrt{41}.

$$x=\frac{\sqrt{41}+9}{2}$$

Nā, me whakaoti te whārite x=\frac{9±\sqrt{41}}{2} ina he tango te ±. Tango \sqrt{41} mai i 9.

$$x=\frac{9-\sqrt{41}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{41}+9}{2} x=\frac{9-\sqrt{41}}{2}$$

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.

$$x^{2}-9x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$x^{2}-9x+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$x^{2}-9x=-10$$

Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-10+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=-10+\frac{81}{4}$$

Tāpiri -10 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{41}{4}$$

Tauwehea x^{2}-9x+\frac{81}{4}. 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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{41}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{\sqrt{41}}{2} x-\frac{9}{2}=-\frac{\sqrt{41}}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{41}+9}{2} x=\frac{9-\sqrt{41}}{2}$$