Whakaoti i te x^2-2x=143

$$x ^ { 2 } - 2 x = 143$$

Whakaoti mō x

$$x=-11$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 143 mai i ngā taha e rua.

$$x^{2}-2x-143=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-2x-143 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.

$$a+b=-2 ab=-143$$

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 -143.

$$1,-143 11,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-143=-142 11-13=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-13 b=11$$

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

$$\left(x-13\right)\left(x+11\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x+11=0.

$$x=13 x=-11$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 143 mai i ngā taha e rua.

$$x^{2}-2x-143=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-143. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-143\right)=-143$$

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 -143.

$$1,-143 11,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-143=-142 11-13=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-13 b=11$$

Tuhia anō te x^{2}-2x-143 hei \left(x^{2}-13x\right)+\left(11x-143\right).

$$\left(x^{2}-13x\right)+\left(11x-143\right)$$

Tauwehea te x i te tuatahi me te 11 i te rōpū tuarua.

$$x\left(x-13\right)+11\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(x+11\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x+11=0.

$$x=13 x=-11$$

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.

$$x^{2}-2x=143$$

Me tango 143 mai i ngā taha e rua o te whārite.

$$x^{2}-2x-143=143-143$$

Mā te tango i te 143 i a ia ake anō ka toe ko te 0.

$$x^{2}-2x-143=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -143 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-143\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-143\right)}}{2}$$

Whakareatia -4 ki te -143.

$$x=\frac{-\left(-2\right)±\sqrt{4+572}}{2}$$

Tāpiri 4 ki te 572.

$$x=\frac{-\left(-2\right)±\sqrt{576}}{2}$$

Tuhia te pūtakerua o te 576.

$$x=\frac{-\left(-2\right)±24}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±24}{2}$$

Nā, me whakaoti te whārite x=\frac{2±24}{2} ina he tāpiri te ±. Tāpiri 2 ki te 24.

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

Whakawehe 26 ki te 2.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{2±24}{2} ina he tango te ±. Tango 24 mai i 2.

$$x=-\frac{22}{2}$$

Whakawehe -22 ki te 2.

$$x=-11$$

Kua oti te whārite te whakatau.

$$x=13 x=-11$$

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.

$$x^{2}-2x=143$$

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

$$x^{2}-2x+1=143+1$$

Tāpiri 143 ki te 1.

$$x^{2}-2x+1=144$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=144$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{144}$$

Whakarūnātia.

$$x-1=12 x-1=-12$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=13 x=-11$$