Whakaoti i te x^2-12x+49=22

$$x ^ { 2 } - 12 x + 49 = 22$$

Whakaoti mō x

$$x=3$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 22 mai i ngā taha e rua.

$$x^{2}-12x+49-22=0$$

Tangohia te 22 i te 49, ka 27.

$$x^{2}-12x+27=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-12x+27 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.

$$a+b=-12 ab=27$$

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 27.

$$-1,-27 -3,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-27=-28 -3-9=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-9 b=-3$$

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

$$\left(x-9\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-3=0.

$$x=9 x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 22 mai i ngā taha e rua.

$$x^{2}-12x+49-22=0$$

Tangohia te 22 i te 49, ka 27.

$$x^{2}-12x+27=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=1\times 27=27$$

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 27.

$$-1,-27 -3,-9$$

Tātaihia te tapeke mō ia takirua.

$$-1-27=-28 -3-9=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-9 b=-3$$

Tuhia anō te x^{2}-12x+27 hei \left(x^{2}-9x\right)+\left(-3x+27\right).

$$\left(x^{2}-9x\right)+\left(-3x+27\right)$$

Tauwehea te x i te tuatahi me te -3 i te rōpū tuarua.

$$x\left(x-9\right)-3\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x-3=0.

$$x=9 x=3$$

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.

$$x^{2}-12x+49=22$$

Me tango 22 mai i ngā taha e rua o te whārite.

$$x^{2}-12x+49-22=22-22$$

Mā te tango i te 22 i a ia ake anō ka toe ko te 0.

$$x^{2}-12x+49-22=0$$

Tango 22 mai i 49.

$$x^{2}-12x+27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me 27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 27}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 27}}{2}$$

Whakareatia -4 ki te 27.

$$x=\frac{-\left(-12\right)±\sqrt{144-108}}{2}$$

Tāpiri 144 ki te -108.

$$x=\frac{-\left(-12\right)±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-12\right)±6}{2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6}{2}$$

Nā, me whakaoti te whārite x=\frac{12±6}{2} ina he tāpiri te ±. Tāpiri 12 ki te 6.

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

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{12±6}{2} ina he tango te ±. Tango 6 mai i 12.

$$x=\frac{6}{2}$$

Whakawehe 6 ki te 2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=9 x=3$$

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.

$$x^{2}-12x+49=22$$

Me tango 49 mai i ngā taha e rua o te whārite.

$$x^{2}-12x+49-49=22-49$$

Mā te tango i te 49 i a ia ake anō ka toe ko te 0.

$$x^{2}-12x=22-49$$

Tango 49 mai i 22.

$$x^{2}-12x=-27$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-27+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-27+36$$

Tāpiri -27 ki te 36.

$$x^{2}-12x+36=9$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=9$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-6=3 x-6=-3$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=9 x=3$$