Whakaoti i te x^2+10x+14=0

$$x ^ { 2 } + 10 x + 14 = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{11}-5\approx -1.68337521$$

$$x=-\left(\sqrt{11}+5\right)\approx -8.31662479$$

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.

$$x^{2}+10x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 10 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 14}}{2}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 14}}{2}$$

Whakareatia -4 ki te 14.

$$x=\frac{-10±\sqrt{100-56}}{2}$$

Tāpiri 100 ki te -56.

$$x=\frac{-10±\sqrt{44}}{2}$$

Tuhia te pūtakerua o te 44.

$$x=\frac{-10±2\sqrt{11}}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}}{2} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{11}.

$$x=\frac{2\sqrt{11}-10}{2}$$

Whakawehe -10+2\sqrt{11} ki te 2.

$$x=\sqrt{11}-5$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}}{2} ina he tango te ±. Tango 2\sqrt{11} mai i -10.

$$x=\frac{-2\sqrt{11}-10}{2}$$

Whakawehe -10-2\sqrt{11} ki te 2.

$$x=-\sqrt{11}-5$$

Kua oti te whārite te whakatau.

$$x=\sqrt{11}-5 x=-\sqrt{11}-5$$

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.

$$x^{2}+10x+14=0$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$x^{2}+10x+14-14=-14$$

Mā te tango i te 14 i a ia ake anō ka toe ko te 0.

$$x^{2}+10x=-14$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=-14+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-14+25$$

Tāpiri -14 ki te 25.

$$x^{2}+10x+25=11$$

Tauwehea x^{2}+10x+25. 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}.

$$\left(x+5\right)^{2}=11$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{11}$$

Whakarūnātia.

$$x+5=\sqrt{11} x+5=-\sqrt{11}$$

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

$$x=\sqrt{11}-5 x=-\sqrt{11}-5$$

Whakaoti mō x

$$x=\sqrt{11}-5\approx -1.68337521$$

$$x=-\sqrt{11}-5\approx -8.31662479$$

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.

$$x^{2}+10x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 10 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 14}}{2}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 14}}{2}$$

Whakareatia -4 ki te 14.

$$x=\frac{-10±\sqrt{100-56}}{2}$$

Tāpiri 100 ki te -56.

$$x=\frac{-10±\sqrt{44}}{2}$$

Tuhia te pūtakerua o te 44.

$$x=\frac{-10±2\sqrt{11}}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}}{2} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{11}.

$$x=\frac{2\sqrt{11}-10}{2}$$

Whakawehe -10+2\sqrt{11} ki te 2.

$$x=\sqrt{11}-5$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{11}}{2} ina he tango te ±. Tango 2\sqrt{11} mai i -10.

$$x=\frac{-2\sqrt{11}-10}{2}$$

Whakawehe -10-2\sqrt{11} ki te 2.

$$x=-\sqrt{11}-5$$

Kua oti te whārite te whakatau.

$$x=\sqrt{11}-5 x=-\sqrt{11}-5$$

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.

$$x^{2}+10x+14=0$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$x^{2}+10x+14-14=-14$$

Mā te tango i te 14 i a ia ake anō ka toe ko te 0.

$$x^{2}+10x=-14$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=-14+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-14+25$$

Tāpiri -14 ki te 25.

$$x^{2}+10x+25=11$$

Tauwehea x^{2}+10x+25. 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}.

$$\left(x+5\right)^{2}=11$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{11}$$

Whakarūnātia.

$$x+5=\sqrt{11} x+5=-\sqrt{11}$$

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

$$x=\sqrt{11}-5 x=-\sqrt{11}-5$$