Whakaoti i te x^2-26x-155=0

$$x ^ { 2 } - 26 x - 155 = 0$$

Whakaoti mō x

$$x=-5$$

$$x=31$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-26x-155 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.

$$a+b=-26 ab=-155$$

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 -155.

$$1,-155 5,-31$$

Tātaihia te tapeke mō ia takirua.

$$1-155=-154 5-31=-26$$

Ko te otinga te takirua ka hoatu i te tapeke -26.

$$a=-31 b=5$$

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

$$\left(x-31\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-31=0 me te x+5=0.

$$x=31 x=-5$$

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-155. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-26 ab=1\left(-155\right)=-155$$

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 -155.

$$1,-155 5,-31$$

Tātaihia te tapeke mō ia takirua.

$$1-155=-154 5-31=-26$$

Ko te otinga te takirua ka hoatu i te tapeke -26.

$$a=-31 b=5$$

Tuhia anō te x^{2}-26x-155 hei \left(x^{2}-31x\right)+\left(5x-155\right).

$$\left(x^{2}-31x\right)+\left(5x-155\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x-31\right)+5\left(x-31\right)$$

Whakatauwehea atu te kīanga pātahi x-31 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-31\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-31=0 me te x+5=0.

$$x=31 x=-5$$

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.

$$x^{2}-26x-155=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -26 mō b, me -155 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-155\right)}}{2}$$

Pūrua -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\left(-155\right)}}{2}$$

Whakareatia -4 ki te -155.

$$x=\frac{-\left(-26\right)±\sqrt{676+620}}{2}$$

Tāpiri 676 ki te 620.

$$x=\frac{-\left(-26\right)±\sqrt{1296}}{2}$$

Tuhia te pūtakerua o te 1296.

$$x=\frac{-\left(-26\right)±36}{2}$$

Ko te tauaro o -26 ko 26.

$$x=\frac{26±36}{2}$$

Nā, me whakaoti te whārite x=\frac{26±36}{2} ina he tāpiri te ±. Tāpiri 26 ki te 36.

$$x=\frac{62}{2}$$

Whakawehe 62 ki te 2.

$$x=31$$

Nā, me whakaoti te whārite x=\frac{26±36}{2} ina he tango te ±. Tango 36 mai i 26.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=31 x=-5$$

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.

$$x^{2}-26x-155=0$$

Me tāpiri 155 ki ngā taha e rua o te whārite.

$$x^{2}-26x-155-\left(-155\right)=-\left(-155\right)$$

Mā te tango i te -155 i a ia ake anō ka toe ko te 0.

$$x^{2}-26x=-\left(-155\right)$$

Tango -155 mai i 0.

$$x^{2}-26x=155$$

Whakawehea te -26, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -13. Nā, tāpiria te pūrua o te -13 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-26x+\left(-13\right)^{2}=155+\left(-13\right)^{2}$$

Pūrua -13.

$$x^{2}-26x+169=155+169$$

Tāpiri 155 ki te 169.

$$x^{2}-26x+169=324$$

Tauwehea x^{2}-26x+169. 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}.

$$\left(x-13\right)^{2}=324$$

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

$$\sqrt{\left(x-13\right)^{2}}=\sqrt{324}$$

Whakarūnātia.

$$x-13=18 x-13=-18$$

Me tāpiri 13 ki ngā taha e rua o te whārite.

$$x=31 x=-5$$