Whakaoti i te x^2+18x+77=0

$${ x }^{ 2 } +18x+77=0$$

Whakaoti mō x

$$x=-11$$

$$x=-7$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+18x+77 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.

$$a+b=18 ab=77$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 77.

$$1,77 7,11$$

Tātaihia te tapeke mō ia takirua.

$$1+77=78 7+11=18$$

Ko te otinga te takirua ka hoatu i te tapeke 18.

$$a=7 b=11$$

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

$$\left(x+7\right)\left(x+11\right)$$

Hei kimi otinga whārite, me whakaoti te x+7=0 me te x+11=0.

$$x=-7 x=-11$$

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+77. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=18 ab=1\times 77=77$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 77.

$$1,77 7,11$$

Tātaihia te tapeke mō ia takirua.

$$1+77=78 7+11=18$$

Ko te otinga te takirua ka hoatu i te tapeke 18.

$$a=7 b=11$$

Tuhia anō te x^{2}+18x+77 hei \left(x^{2}+7x\right)+\left(11x+77\right).

$$\left(x^{2}+7x\right)+\left(11x+77\right)$$

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

$$x\left(x+7\right)+11\left(x+7\right)$$

Whakatauwehea atu te kīanga pātahi x+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+7\right)\left(x+11\right)$$

Hei kimi otinga whārite, me whakaoti te x+7=0 me te x+11=0.

$$x=-7 x=-11$$

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.

$$x^{2}+18x+77=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 77 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 77}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 77}}{2}$$

Whakareatia -4 ki te 77.

$$x=\frac{-18±\sqrt{324-308}}{2}$$

Tāpiri 324 ki te -308.

$$x=\frac{-18±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-18±4}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±4}{2} ina he tāpiri te ±. Tāpiri -18 ki te 4.

$$x=-\frac{14}{2}$$

Whakawehe -14 ki te 2.

$$x=-7$$

Nā, me whakaoti te whārite x=\frac{-18±4}{2} ina he tango te ±. Tango 4 mai i -18.

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

Whakawehe -22 ki te 2.

$$x=-11$$

Kua oti te whārite te whakatau.

$$x=-7 x=-11$$

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.

$$x^{2}+18x+77=0$$

Me tango 77 mai i ngā taha e rua o te whārite.

$$x^{2}+18x+77-77=-77$$

Mā te tango i te 77 i a ia ake anō ka toe ko te 0.

$$x^{2}+18x=-77$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=-77+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-77+81$$

Tāpiri -77 ki te 81.

$$x^{2}+18x+81=4$$

Tauwehea x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=4$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x+9=2 x+9=-2$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=-7 x=-11$$