Whakaoti i te x^2-8x+9=0

$$x ^ { 2 } - 8 x + 9 = 0$$

Whakaoti mō x

$$x=\sqrt{7}+4\approx 6.645751311$$

$$x=4-\sqrt{7}\approx 1.354248689$$

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.

$$x^{2}-8x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 9}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 9}}{2}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-8\right)±\sqrt{64-36}}{2}$$

Tāpiri 64 ki te -36.

$$x=\frac{-\left(-8\right)±\sqrt{28}}{2}$$

Tuhia te pūtakerua o te 28.

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

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{7}}{2}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{7}}{2} ina he tāpiri te ±. Tāpiri 8 ki te 2\sqrt{7}.

$$x=\frac{2\sqrt{7}+8}{2}$$

Whakawehe 8+2\sqrt{7} ki te 2.

$$x=\sqrt{7}+4$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{7}}{2} ina he tango te ±. Tango 2\sqrt{7} mai i 8.

$$x=\frac{8-2\sqrt{7}}{2}$$

Whakawehe 8-2\sqrt{7} ki te 2.

$$x=4-\sqrt{7}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{7}+4 x=4-\sqrt{7}$$

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.

$$x^{2}-8x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x^{2}-8x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$x^{2}-8x=-9$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=-9+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-9+16$$

Tāpiri -9 ki te 16.

$$x^{2}-8x+16=7$$

Tauwehea x^{2}-8x+16. 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}.

$$\left(x-4\right)^{2}=7$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{7}$$

Whakarūnātia.

$$x-4=\sqrt{7} x-4=-\sqrt{7}$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=\sqrt{7}+4 x=4-\sqrt{7}$$