Whakaoti i te z^2+14=9z

$$z ^ { 2 } + 14 = 9 z$$

Whakaoti mō z

$$z=2$$

$$z=7$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 9z mai i ngā taha e rua.

$$z^{2}+14-9z=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$z^{2}-9z+14=0$$

Hei whakaoti i te whārite, whakatauwehea te z^{2}-9z+14 mā te whakamahi i te tātai z^{2}+\left(a+b\right)z+ab=\left(z+a\right)\left(z+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 tōraro te a+b, he tōraro 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.

$$a=-7 b=-2$$

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

$$\left(z-7\right)\left(z-2\right)$$

Hei kimi otinga whārite, me whakaoti te z-7=0 me te z-2=0.

$$z=7 z=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 9z mai i ngā taha e rua.

$$z^{2}+14-9z=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$z^{2}-9z+14=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei z^{2}+az+bz+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 tōraro te a+b, he tōraro 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.

$$a=-7 b=-2$$

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

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

Tauwehea te z i te tuatahi me te -2 i te rōpū tuarua.

$$z\left(z-7\right)-2\left(z-7\right)$$

Whakatauwehea atu te kīanga pātahi z-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(z-7\right)\left(z-2\right)$$

Hei kimi otinga whārite, me whakaoti te z-7=0 me te z-2=0.

$$z=7 z=2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 9z mai i ngā taha e rua.

$$z^{2}+14-9z=0$$

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.

$$z^{2}-9z+14=0$$

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}.

$$z=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 14}}{2}$$

Pūrua -9.

$$z=\frac{-\left(-9\right)±\sqrt{81-4\times 14}}{2}$$

Whakareatia -4 ki te 14.

$$z=\frac{-\left(-9\right)±\sqrt{81-56}}{2}$$

Tāpiri 81 ki te -56.

$$z=\frac{-\left(-9\right)±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$z=\frac{-\left(-9\right)±5}{2}$$

Ko te tauaro o -9 ko 9.

$$z=\frac{9±5}{2}$$

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

$$z=\frac{14}{2}$$

Whakawehe 14 ki te 2.

$$z=7$$

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

$$z=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$z=2$$

Kua oti te whārite te whakatau.

$$z=7 z=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 9z mai i ngā taha e rua.

$$z^{2}+14-9z=0$$

Tangohia te 14 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$z^{2}-9z=-14$$

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.

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

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

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

Tāpiri -14 ki te \frac{81}{4}.

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

Tauwehea z^{2}-9z+\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}.

$$\left(z-\frac{9}{2}\right)^{2}=\frac{25}{4}$$

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

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

Whakarūnātia.

$$z-\frac{9}{2}=\frac{5}{2} z-\frac{9}{2}=-\frac{5}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$z=7 z=2$$