Whakaoti i te t^2+11t+24=0

$$t ^ { 2 } + 11 t + 24 = 0$$

Whakaoti mō t

$$t=-8$$

$$t=-3$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te t^{2}+11t+24 mā te whakamahi i te tātai t^{2}+\left(a+b\right)t+ab=\left(t+a\right)\left(t+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Tātaihia te tapeke mō ia takirua.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=3 b=8$$

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

$$\left(t+3\right)\left(t+8\right)$$

Hei kimi otinga whārite, me whakaoti te t+3=0 me te t+8=0.

$$t=-3 t=-8$$

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 t^{2}+at+bt+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=1\times 24=24$$

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 24.

$$1,24 2,12 3,8 4,6$$

Tātaihia te tapeke mō ia takirua.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=3 b=8$$

Tuhia anō te t^{2}+11t+24 hei \left(t^{2}+3t\right)+\left(8t+24\right).

$$\left(t^{2}+3t\right)+\left(8t+24\right)$$

Tauwehea te t i te tuatahi me te 8 i te rōpū tuarua.

$$t\left(t+3\right)+8\left(t+3\right)$$

Whakatauwehea atu te kīanga pātahi t+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t+3\right)\left(t+8\right)$$

Hei kimi otinga whārite, me whakaoti te t+3=0 me te t+8=0.

$$t=-3 t=-8$$

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.

$$t^{2}+11t+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 11 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-11±\sqrt{11^{2}-4\times 24}}{2}$$

Pūrua 11.

$$t=\frac{-11±\sqrt{121-4\times 24}}{2}$$

Whakareatia -4 ki te 24.

$$t=\frac{-11±\sqrt{121-96}}{2}$$

Tāpiri 121 ki te -96.

$$t=\frac{-11±\sqrt{25}}{2}$$

Tuhia te pūtakerua o te 25.

$$t=\frac{-11±5}{2}$$

Nā, me whakaoti te whārite t=\frac{-11±5}{2} ina he tāpiri te ±. Tāpiri -11 ki te 5.

$$t=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$t=-3$$

Nā, me whakaoti te whārite t=\frac{-11±5}{2} ina he tango te ±. Tango 5 mai i -11.

$$t=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$t=-8$$

Kua oti te whārite te whakatau.

$$t=-3 t=-8$$

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.

$$t^{2}+11t+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$t^{2}+11t+24-24=-24$$

Mā te tango i te 24 i a ia ake anō ka toe ko te 0.

$$t^{2}+11t=-24$$

Whakawehea te 11, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{2}. Nā, tāpiria te pūrua o te \frac{11}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+11t+\left(\frac{11}{2}\right)^{2}=-24+\left(\frac{11}{2}\right)^{2}$$

Pūruatia \frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+11t+\frac{121}{4}=-24+\frac{121}{4}$$

Tāpiri -24 ki te \frac{121}{4}.

$$t^{2}+11t+\frac{121}{4}=\frac{25}{4}$$

Tauwehea t^{2}+11t+\frac{121}{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}.

$$\left(t+\frac{11}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(t+\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$t+\frac{11}{2}=\frac{5}{2} t+\frac{11}{2}=-\frac{5}{2}$$

Me tango \frac{11}{2} mai i ngā taha e rua o te whārite.

$$t=-3 t=-8$$