Whakaoti i te x^2-8x-1029=0=

$${ x }^{ 2 } -8x-1029=0=$$

Whakaoti mō x

$$x=\sqrt{1045}+4\approx 36.32645975$$

$$x=4-\sqrt{1045}\approx -28.32645975$$

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.

$$x^{2}-8x-1029=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me -1029 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1029\right)}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1029\right)}}{2}$$

Whakareatia -4 ki te -1029.

$$x=\frac{-\left(-8\right)±\sqrt{64+4116}}{2}$$

Tāpiri 64 ki te 4116.

$$x=\frac{-\left(-8\right)±\sqrt{4180}}{2}$$

Tuhia te pūtakerua o te 4180.

$$x=\frac{-\left(-8\right)±2\sqrt{1045}}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{1045}}{2}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{1045}}{2} ina he tāpiri te ±. Tāpiri 8 ki te 2\sqrt{1045}.

$$x=\frac{2\sqrt{1045}+8}{2}$$

Whakawehe 8+2\sqrt{1045} ki te 2.

$$x=\sqrt{1045}+4$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{1045}}{2} ina he tango te ±. Tango 2\sqrt{1045} mai i 8.

$$x=\frac{8-2\sqrt{1045}}{2}$$

Whakawehe 8-2\sqrt{1045} ki te 2.

$$x=4-\sqrt{1045}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{1045}+4 x=4-\sqrt{1045}$$

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.

$$x^{2}-8x-1029=0$$

Me tāpiri 1029 ki ngā taha e rua o te whārite.

$$x^{2}-8x-1029-\left(-1029\right)=-\left(-1029\right)$$

Mā te tango i te -1029 i a ia ake anō ka toe ko te 0.

$$x^{2}-8x=-\left(-1029\right)$$

Tango -1029 mai i 0.

$$x^{2}-8x=1029$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=1029+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=1029+16$$

Tāpiri 1029 ki te 16.

$$x^{2}-8x+16=1045$$

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}.

$$\left(x-4\right)^{2}=1045$$

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

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{1045}$$

Whakarūnātia.

$$x-4=\sqrt{1045} x-4=-\sqrt{1045}$$

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

$$x=\sqrt{1045}+4 x=4-\sqrt{1045}$$