Whakaoti i te x^2-14x=-47

$$x ^ { 2 } - 14 x = - 47$$

Whakaoti mō x

$$x=\sqrt{2}+7\approx 8.414213562$$

$$x=7-\sqrt{2}\approx 5.585786438$$

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.

$$x^{2}-14x=-47$$

Me tāpiri 47 ki ngā taha e rua o te whārite.

$$x^{2}-14x-\left(-47\right)=-47-\left(-47\right)$$

Mā te tango i te -47 i a ia ake anō ka toe ko te 0.

$$x^{2}-14x-\left(-47\right)=0$$

Tango -47 mai i 0.

$$x^{2}-14x+47=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 47 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 47}}{2}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 47}}{2}$$

Whakareatia -4 ki te 47.

$$x=\frac{-\left(-14\right)±\sqrt{196-188}}{2}$$

Tāpiri 196 ki te -188.

$$x=\frac{-\left(-14\right)±\sqrt{8}}{2}$$

Tuhia te pūtakerua o te 8.

$$x=\frac{-\left(-14\right)±2\sqrt{2}}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±2\sqrt{2}}{2}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri 14 ki te 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+14}{2}$$

Whakawehe 14+2\sqrt{2} ki te 2.

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

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{2}}{2} ina he tango te ±. Tango 2\sqrt{2} mai i 14.

$$x=\frac{14-2\sqrt{2}}{2}$$

Whakawehe 14-2\sqrt{2} ki te 2.

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

Kua oti te whārite te whakatau.

$$x=\sqrt{2}+7 x=7-\sqrt{2}$$

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.

$$x^{2}-14x=-47$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-47+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-47+49$$

Tāpiri -47 ki te 49.

$$x^{2}-14x+49=2$$

Tauwehea x^{2}-14x+49. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=\sqrt{2}+7 x=7-\sqrt{2}$$