Whakaoti i te t^2-109t+900=0

$${ t }^{ 2 } -109t+900 = 0$$

Whakaoti mō t

$$t=9$$

$$t=100$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te t^{2}-109t+900 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.

$$a+b=-109 ab=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Tātaihia te tapeke mō ia takirua.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

Ko te otinga te takirua ka hoatu i te tapeke -109.

$$a=-100 b=-9$$

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

$$\left(t-100\right)\left(t-9\right)$$

Hei kimi otinga whārite, me whakaoti te t-100=0 me te t-9=0.

$$t=100 t=9$$

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+900. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-109 ab=1\times 900=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Tātaihia te tapeke mō ia takirua.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

Ko te otinga te takirua ka hoatu i te tapeke -109.

$$a=-100 b=-9$$

Tuhia anō te t^{2}-109t+900 hei \left(t^{2}-100t\right)+\left(-9t+900\right).

$$\left(t^{2}-100t\right)+\left(-9t+900\right)$$

Tauwehea te t i te tuatahi me te -9 i te rōpū tuarua.

$$t\left(t-100\right)-9\left(t-100\right)$$

Whakatauwehea atu te kīanga pātahi t-100 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-100\right)\left(t-9\right)$$

Hei kimi otinga whārite, me whakaoti te t-100=0 me te t-9=0.

$$t=100 t=9$$

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.

$$t^{2}-109t+900=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -109 mō b, me 900 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-109\right)±\sqrt{\left(-109\right)^{2}-4\times 900}}{2}$$

Pūrua -109.

$$t=\frac{-\left(-109\right)±\sqrt{11881-4\times 900}}{2}$$

Whakareatia -4 ki te 900.

$$t=\frac{-\left(-109\right)±\sqrt{11881-3600}}{2}$$

Tāpiri 11881 ki te -3600.

$$t=\frac{-\left(-109\right)±\sqrt{8281}}{2}$$

Tuhia te pūtakerua o te 8281.

$$t=\frac{-\left(-109\right)±91}{2}$$

Ko te tauaro o -109 ko 109.

$$t=\frac{109±91}{2}$$

Nā, me whakaoti te whārite t=\frac{109±91}{2} ina he tāpiri te ±. Tāpiri 109 ki te 91.

$$t=\frac{200}{2}$$

Whakawehe 200 ki te 2.

$$t=100$$

Nā, me whakaoti te whārite t=\frac{109±91}{2} ina he tango te ±. Tango 91 mai i 109.

$$t=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$t=9$$

Kua oti te whārite te whakatau.

$$t=100 t=9$$

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.

$$t^{2}-109t+900=0$$

Me tango 900 mai i ngā taha e rua o te whārite.

$$t^{2}-109t+900-900=-900$$

Mā te tango i te 900 i a ia ake anō ka toe ko te 0.

$$t^{2}-109t=-900$$

Whakawehea te -109, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{109}{2}. Nā, tāpiria te pūrua o te -\frac{109}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-109t+\left(-\frac{109}{2}\right)^{2}=-900+\left(-\frac{109}{2}\right)^{2}$$

Pūruatia -\frac{109}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-109t+\frac{11881}{4}=-900+\frac{11881}{4}$$

Tāpiri -900 ki te \frac{11881}{4}.

$$t^{2}-109t+\frac{11881}{4}=\frac{8281}{4}$$

Tauwehea t^{2}-109t+\frac{11881}{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}.

$$\left(t-\frac{109}{2}\right)^{2}=\frac{8281}{4}$$

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

$$\sqrt{\left(t-\frac{109}{2}\right)^{2}}=\sqrt{\frac{8281}{4}}$$

Whakarūnātia.

$$t-\frac{109}{2}=\frac{91}{2} t-\frac{109}{2}=-\frac{91}{2}$$

Me tāpiri \frac{109}{2} ki ngā taha e rua o te whārite.

$$t=100 t=9$$