Whakaoti i te t^2-t=992

$$t ^ { 2 } - t = 992$$

Whakaoti mō t

$$t=-31$$

$$t=32$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 992 mai i ngā taha e rua.

$$t^{2}-t-992=0$$

Hei whakaoti i te whārite, whakatauwehea te t^{2}-t-992 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.

$$a+b=-1 ab=-992$$

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 -992.

$$1,-992 2,-496 4,-248 8,-124 16,-62 31,-32$$

Tātaihia te tapeke mō ia takirua.

$$1-992=-991 2-496=-494 4-248=-244 8-124=-116 16-62=-46 31-32=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-32 b=31$$

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

$$\left(t-32\right)\left(t+31\right)$$

Hei kimi otinga whārite, me whakaoti te t-32=0 me te t+31=0.

$$t=32 t=-31$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 992 mai i ngā taha e rua.

$$t^{2}-t-992=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei t^{2}+at+bt-992. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-992\right)=-992$$

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 -992.

$$1,-992 2,-496 4,-248 8,-124 16,-62 31,-32$$

Tātaihia te tapeke mō ia takirua.

$$1-992=-991 2-496=-494 4-248=-244 8-124=-116 16-62=-46 31-32=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-32 b=31$$

Tuhia anō te t^{2}-t-992 hei \left(t^{2}-32t\right)+\left(31t-992\right).

$$\left(t^{2}-32t\right)+\left(31t-992\right)$$

Tauwehea te t i te tuatahi me te 31 i te rōpū tuarua.

$$t\left(t-32\right)+31\left(t-32\right)$$

Whakatauwehea atu te kīanga pātahi t-32 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-32\right)\left(t+31\right)$$

Hei kimi otinga whārite, me whakaoti te t-32=0 me te t+31=0.

$$t=32 t=-31$$

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.

$$t^{2}-t=992$$

Me tango 992 mai i ngā taha e rua o te whārite.

$$t^{2}-t-992=992-992$$

Mā te tango i te 992 i a ia ake anō ka toe ko te 0.

$$t^{2}-t-992=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -992 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-1\right)±\sqrt{1-4\left(-992\right)}}{2}$$

Whakareatia -4 ki te -992.

$$t=\frac{-\left(-1\right)±\sqrt{1+3968}}{2}$$

Tāpiri 1 ki te 3968.

$$t=\frac{-\left(-1\right)±\sqrt{3969}}{2}$$

Tuhia te pūtakerua o te 3969.

$$t=\frac{-\left(-1\right)±63}{2}$$

Ko te tauaro o -1 ko 1.

$$t=\frac{1±63}{2}$$

Nā, me whakaoti te whārite t=\frac{1±63}{2} ina he tāpiri te ±. Tāpiri 1 ki te 63.

$$t=\frac{64}{2}$$

Whakawehe 64 ki te 2.

$$t=32$$

Nā, me whakaoti te whārite t=\frac{1±63}{2} ina he tango te ±. Tango 63 mai i 1.

$$t=-\frac{62}{2}$$

Whakawehe -62 ki te 2.

$$t=-31$$

Kua oti te whārite te whakatau.

$$t=32 t=-31$$

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.

$$t^{2}-t=992$$

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-t+\left(-\frac{1}{2}\right)^{2}=992+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-t+\frac{1}{4}=992+\frac{1}{4}$$

Tāpiri 992 ki te \frac{1}{4}.

$$t^{2}-t+\frac{1}{4}=\frac{3969}{4}$$

Tauwehea t^{2}-t+\frac{1}{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}.

$$\left(t-\frac{1}{2}\right)^{2}=\frac{3969}{4}$$

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

$$\sqrt{\left(t-\frac{1}{2}\right)^{2}}=\sqrt{\frac{3969}{4}}$$

Whakarūnātia.

$$t-\frac{1}{2}=\frac{63}{2} t-\frac{1}{2}=-\frac{63}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$t=32 t=-31$$