Whakaoti i te x^2-16x+57=0

$$x ^ { 2 } - 16 x + 57 = 0$$

Whakaoti mō x

$$x=\sqrt{7}+8\approx 10.645751311$$

$$x=8-\sqrt{7}\approx 5.354248689$$

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.

$$x^{2}-16x+57=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -16 mō b, me 57 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 57}}{2}$$

Pūrua -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 57}}{2}$$

Whakareatia -4 ki te 57.

$$x=\frac{-\left(-16\right)±\sqrt{256-228}}{2}$$

Tāpiri 256 ki te -228.

$$x=\frac{-\left(-16\right)±\sqrt{28}}{2}$$

Tuhia te pūtakerua o te 28.

$$x=\frac{-\left(-16\right)±2\sqrt{7}}{2}$$

Ko te tauaro o -16 ko 16.

$$x=\frac{16±2\sqrt{7}}{2}$$

Nā, me whakaoti te whārite x=\frac{16±2\sqrt{7}}{2} ina he tāpiri te ±. Tāpiri 16 ki te 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+16}{2}$$

Whakawehe 16+2\sqrt{7} ki te 2.

$$x=\sqrt{7}+8$$

Nā, me whakaoti te whārite x=\frac{16±2\sqrt{7}}{2} ina he tango te ±. Tango 2\sqrt{7} mai i 16.

$$x=\frac{16-2\sqrt{7}}{2}$$

Whakawehe 16-2\sqrt{7} ki te 2.

$$x=8-\sqrt{7}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{7}+8 x=8-\sqrt{7}$$

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.

$$x^{2}-16x+57=0$$

Me tango 57 mai i ngā taha e rua o te whārite.

$$x^{2}-16x+57-57=-57$$

Mā te tango i te 57 i a ia ake anō ka toe ko te 0.

$$x^{2}-16x=-57$$

Whakawehea te -16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -8. Nā, tāpiria te pūrua o te -8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-16x+\left(-8\right)^{2}=-57+\left(-8\right)^{2}$$

Pūrua -8.

$$x^{2}-16x+64=-57+64$$

Tāpiri -57 ki te 64.

$$x^{2}-16x+64=7$$

Tauwehea x^{2}-16x+64. 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}.

$$\left(x-8\right)^{2}=7$$

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

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{7}$$

Whakarūnātia.

$$x-8=\sqrt{7} x-8=-\sqrt{7}$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x=\sqrt{7}+8 x=8-\sqrt{7}$$