Whakaoti i te x^2+14x+5=0

$$x ^ { 2 } + 14 x + 5 = 0$$

Whakaoti mō x

$$x=2\sqrt{11}-7\approx -0.366750419$$

$$x=-2\sqrt{11}-7\approx -13.633249581$$

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.

$$x^{2}+14x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 5}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 5}}{2}$$

Whakareatia -4 ki te 5.

$$x=\frac{-14±\sqrt{196-20}}{2}$$

Tāpiri 196 ki te -20.

$$x=\frac{-14±\sqrt{176}}{2}$$

Tuhia te pūtakerua o te 176.

$$x=\frac{-14±4\sqrt{11}}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±4\sqrt{11}}{2} ina he tāpiri te ±. Tāpiri -14 ki te 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-14}{2}$$

Whakawehe -14+4\sqrt{11} ki te 2.

$$x=2\sqrt{11}-7$$

Nā, me whakaoti te whārite x=\frac{-14±4\sqrt{11}}{2} ina he tango te ±. Tango 4\sqrt{11} mai i -14.

$$x=\frac{-4\sqrt{11}-14}{2}$$

Whakawehe -14-4\sqrt{11} ki te 2.

$$x=-2\sqrt{11}-7$$

Kua oti te whārite te whakatau.

$$x=2\sqrt{11}-7 x=-2\sqrt{11}-7$$

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.

$$x^{2}+14x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x^{2}+14x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-5$$

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.

$$x^{2}+14x+7^{2}=-5+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=-5+49$$

Tāpiri -5 ki te 49.

$$x^{2}+14x+49=44$$

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}.

$$\left(x+7\right)^{2}=44$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{44}$$

Whakarūnātia.

$$x+7=2\sqrt{11} x+7=-2\sqrt{11}$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=2\sqrt{11}-7 x=-2\sqrt{11}-7$$