Whakaoti i te m^2-5m-14=0

$$m ^ { 2 } - 5 m - 14 = 0$$

Whakaoti mō m

$$m=-2$$

$$m=7$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te m^{2}-5m-14 mā te whakamahi i te tātai m^{2}+\left(a+b\right)m+ab=\left(m+a\right)\left(m+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=-14$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -14.

$$1,-14 2,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-14=-13 2-7=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-7 b=2$$

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

$$\left(m-7\right)\left(m+2\right)$$

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

$$m=7 m=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$a+b=-5 ab=1\left(-14\right)=-14$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -14.

$$1,-14 2,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-14=-13 2-7=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-7 b=2$$

Tuhia anō te m^{2}-5m-14 hei \left(m^{2}-7m\right)+\left(2m-14\right).

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

Tauwehea te m i te tuatahi me te 2 i te rōpū tuarua.

$$m\left(m-7\right)+2\left(m-7\right)$$

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

$$\left(m-7\right)\left(m+2\right)$$

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

$$m=7 m=-2$$

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.

$$m^{2}-5m-14=0$$

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

$$m=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-14\right)}}{2}$$

Pūrua -5.

$$m=\frac{-\left(-5\right)±\sqrt{25-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$m=\frac{-\left(-5\right)±\sqrt{25+56}}{2}$$

Tāpiri 25 ki te 56.

$$m=\frac{-\left(-5\right)±\sqrt{81}}{2}$$

Tuhia te pūtakerua o te 81.

$$m=\frac{-\left(-5\right)±9}{2}$$

Ko te tauaro o -5 ko 5.

$$m=\frac{5±9}{2}$$

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

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

Whakawehe 14 ki te 2.

$$m=7$$

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

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

Whakawehe -4 ki te 2.

$$m=-2$$

Kua oti te whārite te whakatau.

$$m=7 m=-2$$

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.

$$m^{2}-5m-14=0$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$m^{2}-5m-14-\left(-14\right)=-\left(-14\right)$$

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

$$m^{2}-5m=-\left(-14\right)$$

Tango -14 mai i 0.

$$m^{2}-5m=14$$

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

$$m^{2}-5m+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$m^{2}-5m+\frac{25}{4}=14+\frac{25}{4}$$

Tāpiri 14 ki te \frac{25}{4}.

$$m^{2}-5m+\frac{25}{4}=\frac{81}{4}$$

Tauwehea m^{2}-5m+\frac{25}{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}.

$$\left(m-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(m-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$m=7 m=-2$$