Whakaoti i te x^2-98x-1443=0

$$x ^ { 2 } - 98 x - 1443 = 0$$

Whakaoti mō x

$$x=-13$$

$$x=111$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-98x-1443 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-98 ab=-1443$$

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 -1443.

$$1,-1443 3,-481 13,-111 37,-39$$

Tātaihia te tapeke mō ia takirua.

$$1-1443=-1442 3-481=-478 13-111=-98 37-39=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -98.

$$a=-111 b=13$$

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

$$\left(x-111\right)\left(x+13\right)$$

Hei kimi otinga whārite, me whakaoti te x-111=0 me te x+13=0.

$$x=111 x=-13$$

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 x^{2}+ax+bx-1443. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-98 ab=1\left(-1443\right)=-1443$$

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 -1443.

$$1,-1443 3,-481 13,-111 37,-39$$

Tātaihia te tapeke mō ia takirua.

$$1-1443=-1442 3-481=-478 13-111=-98 37-39=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -98.

$$a=-111 b=13$$

Tuhia anō te x^{2}-98x-1443 hei \left(x^{2}-111x\right)+\left(13x-1443\right).

$$\left(x^{2}-111x\right)+\left(13x-1443\right)$$

Tauwehea te x i te tuatahi me te 13 i te rōpū tuarua.

$$x\left(x-111\right)+13\left(x-111\right)$$

Whakatauwehea atu te kīanga pātahi x-111 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-111\right)\left(x+13\right)$$

Hei kimi otinga whārite, me whakaoti te x-111=0 me te x+13=0.

$$x=111 x=-13$$

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.

$$x^{2}-98x-1443=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -98 mō b, me -1443 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-98\right)±\sqrt{\left(-98\right)^{2}-4\left(-1443\right)}}{2}$$

Pūrua -98.

$$x=\frac{-\left(-98\right)±\sqrt{9604-4\left(-1443\right)}}{2}$$

Whakareatia -4 ki te -1443.

$$x=\frac{-\left(-98\right)±\sqrt{9604+5772}}{2}$$

Tāpiri 9604 ki te 5772.

$$x=\frac{-\left(-98\right)±\sqrt{15376}}{2}$$

Tuhia te pūtakerua o te 15376.

$$x=\frac{-\left(-98\right)±124}{2}$$

Ko te tauaro o -98 ko 98.

$$x=\frac{98±124}{2}$$

Nā, me whakaoti te whārite x=\frac{98±124}{2} ina he tāpiri te ±. Tāpiri 98 ki te 124.

$$x=\frac{222}{2}$$

Whakawehe 222 ki te 2.

$$x=111$$

Nā, me whakaoti te whārite x=\frac{98±124}{2} ina he tango te ±. Tango 124 mai i 98.

$$x=-\frac{26}{2}$$

Whakawehe -26 ki te 2.

$$x=-13$$

Kua oti te whārite te whakatau.

$$x=111 x=-13$$

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.

$$x^{2}-98x-1443=0$$

Me tāpiri 1443 ki ngā taha e rua o te whārite.

$$x^{2}-98x-1443-\left(-1443\right)=-\left(-1443\right)$$

Mā te tango i te -1443 i a ia ake anō ka toe ko te 0.

$$x^{2}-98x=-\left(-1443\right)$$

Tango -1443 mai i 0.

$$x^{2}-98x=1443$$

Whakawehea te -98, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -49. Nā, tāpiria te pūrua o te -49 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-98x+\left(-49\right)^{2}=1443+\left(-49\right)^{2}$$

Pūrua -49.

$$x^{2}-98x+2401=1443+2401$$

Tāpiri 1443 ki te 2401.

$$x^{2}-98x+2401=3844$$

Tauwehea x^{2}-98x+2401. 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}.

$$\left(x-49\right)^{2}=3844$$

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

$$\sqrt{\left(x-49\right)^{2}}=\sqrt{3844}$$

Whakarūnātia.

$$x-49=62 x-49=-62$$

Me tāpiri 49 ki ngā taha e rua o te whārite.

$$x=111 x=-13$$