Whakaoti i te x^2+9x-2=-16

$$x ^ { 2 } + 9 x - 2 = - 16$$

Whakaoti mō x

$$x=-7$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 16 ki ngā taha e rua.

$$x^{2}+9x-2+16=0$$

Tāpirihia te -2 ki te 16, ka 14.

$$x^{2}+9x+14=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+9x+14 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=14$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 14.

$$1,14 2,7$$

Tātaihia te tapeke mō ia takirua.

$$1+14=15 2+7=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=2 b=7$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x+2\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+7=0.

$$x=-2 x=-7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 16 ki ngā taha e rua.

$$x^{2}+9x-2+16=0$$

Tāpirihia te -2 ki te 16, ka 14.

$$x^{2}+9x+14=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=1\times 14=14$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 14.

$$1,14 2,7$$

Tātaihia te tapeke mō ia takirua.

$$1+14=15 2+7=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=2 b=7$$

Tuhia anō te x^{2}+9x+14 hei \left(x^{2}+2x\right)+\left(7x+14\right).

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

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(x+2\right)+7\left(x+2\right)$$

Whakatauwehea atu te kīanga pātahi x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+2\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+7=0.

$$x=-2 x=-7$$

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.

$$x^{2}+9x-2=-16$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$x^{2}+9x-2-\left(-16\right)=-16-\left(-16\right)$$

Mā te tango i te -16 i a ia ake anō ka toe ko te 0.

$$x^{2}+9x-2-\left(-16\right)=0$$

Tango -16 mai i -2.

$$x^{2}+9x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 9 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 14}}{2}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 14}}{2}$$

Whakareatia -4 ki te 14.

$$x=\frac{-9±\sqrt{81-56}}{2}$$

Tāpiri 81 ki te -56.

$$x=\frac{-9±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-9±5}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±5}{2} ina he tāpiri te ±. Tāpiri -9 ki te 5.

$$x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-9±5}{2} ina he tango te ±. Tango 5 mai i -9.

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

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=-2 x=-7$$

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.

$$x^{2}+9x-2=-16$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x^{2}+9x-2-\left(-2\right)=-16-\left(-2\right)$$

Mā te tango i te -2 i a ia ake anō ka toe ko te 0.

$$x^{2}+9x=-16-\left(-2\right)$$

Tango -2 mai i -16.

$$x^{2}+9x=-14$$

Whakawehea te 9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{2}. Nā, tāpiria te pūrua o te \frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-14+\left(\frac{9}{2}\right)^{2}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+9x+\frac{81}{4}=-14+\frac{81}{4}$$

Tāpiri -14 ki te \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{25}{4}$$

Tauwehea x^{2}+9x+\frac{81}{4}. 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}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x+\frac{9}{2}=\frac{5}{2} x+\frac{9}{2}=-\frac{5}{2}$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=-2 x=-7$$