Whakaoti i te x^2+14x+32=0

$$x ^ { 2 } + 14 x + 32 = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{17}-7\approx -2.876894374$$

$$x=-\left(\sqrt{17}+7\right)\approx -11.123105626$$

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.

$$x^{2}+14x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 32}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 32}}{2}$$

Whakareatia -4 ki te 32.

$$x=\frac{-14±\sqrt{196-128}}{2}$$

Tāpiri 196 ki te -128.

$$x=\frac{-14±\sqrt{68}}{2}$$

Tuhia te pūtakerua o te 68.

$$x=\frac{-14±2\sqrt{17}}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{17}}{2} ina he tāpiri te ±. Tāpiri -14 ki te 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-14}{2}$$

Whakawehe -14+2\sqrt{17} ki te 2.

$$x=\sqrt{17}-7$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{17}}{2} ina he tango te ±. Tango 2\sqrt{17} mai i -14.

$$x=\frac{-2\sqrt{17}-14}{2}$$

Whakawehe -14-2\sqrt{17} ki te 2.

$$x=-\sqrt{17}-7$$

Kua oti te whārite te whakatau.

$$x=\sqrt{17}-7 x=-\sqrt{17}-7$$

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.

$$x^{2}+14x+32=0$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$x^{2}+14x+32-32=-32$$

Mā te tango i te 32 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-32$$

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.

$$x^{2}+14x+7^{2}=-32+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=-32+49$$

Tāpiri -32 ki te 49.

$$x^{2}+14x+49=17$$

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}.

$$\left(x+7\right)^{2}=17$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{17}$$

Whakarūnātia.

$$x+7=\sqrt{17} x+7=-\sqrt{17}$$

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

$$x=\sqrt{17}-7 x=-\sqrt{17}-7$$

Whakaoti mō x

$$x=\sqrt{17}-7\approx -2.876894374$$

$$x=-\sqrt{17}-7\approx -11.123105626$$

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.

$$x^{2}+14x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 32}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 32}}{2}$$

Whakareatia -4 ki te 32.

$$x=\frac{-14±\sqrt{196-128}}{2}$$

Tāpiri 196 ki te -128.

$$x=\frac{-14±\sqrt{68}}{2}$$

Tuhia te pūtakerua o te 68.

$$x=\frac{-14±2\sqrt{17}}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{17}}{2} ina he tāpiri te ±. Tāpiri -14 ki te 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-14}{2}$$

Whakawehe -14+2\sqrt{17} ki te 2.

$$x=\sqrt{17}-7$$

Nā, me whakaoti te whārite x=\frac{-14±2\sqrt{17}}{2} ina he tango te ±. Tango 2\sqrt{17} mai i -14.

$$x=\frac{-2\sqrt{17}-14}{2}$$

Whakawehe -14-2\sqrt{17} ki te 2.

$$x=-\sqrt{17}-7$$

Kua oti te whārite te whakatau.

$$x=\sqrt{17}-7 x=-\sqrt{17}-7$$

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.

$$x^{2}+14x+32=0$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$x^{2}+14x+32-32=-32$$

Mā te tango i te 32 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-32$$

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.

$$x^{2}+14x+7^{2}=-32+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=-32+49$$

Tāpiri -32 ki te 49.

$$x^{2}+14x+49=17$$

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}.

$$\left(x+7\right)^{2}=17$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{17}$$

Whakarūnātia.

$$x+7=\sqrt{17} x+7=-\sqrt{17}$$

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

$$x=\sqrt{17}-7 x=-\sqrt{17}-7$$