Whakaoti i te a^2+33-14a=0

$$a ^ { 2 } + 33 - 14 a = 0$$

Whakaoti mō a

$$a=3$$

$$a=11$$

Ngā Upane Whakamahi Tauwehe

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$a^{2}-14a+33=0$$

Hei whakaoti i te whārite, whakatauwehea te a^{2}-14a+33 mā te whakamahi i te tātai a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=33$$

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 33.

$$-1,-33 -3,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-33=-34 -3-11=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-11 b=-3$$

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

$$\left(a-11\right)\left(a-3\right)$$

Hei kimi otinga whārite, me whakaoti te a-11=0 me te a-3=0.

$$a=11 a=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$a^{2}-14a+33=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei a^{2}+aa+ba+33. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=1\times 33=33$$

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 33.

$$-1,-33 -3,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-33=-34 -3-11=-14$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-11 b=-3$$

Tuhia anō te a^{2}-14a+33 hei \left(a^{2}-11a\right)+\left(-3a+33\right).

$$\left(a^{2}-11a\right)+\left(-3a+33\right)$$

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

$$a\left(a-11\right)-3\left(a-11\right)$$

Whakatauwehea atu te kīanga pātahi a-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a-11\right)\left(a-3\right)$$

Hei kimi otinga whārite, me whakaoti te a-11=0 me te a-3=0.

$$a=11 a=3$$

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.

$$a^{2}-14a+33=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -14 mō b, me 33 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 33}}{2}$$

Pūrua -14.

$$a=\frac{-\left(-14\right)±\sqrt{196-4\times 33}}{2}$$

Whakareatia -4 ki te 33.

$$a=\frac{-\left(-14\right)±\sqrt{196-132}}{2}$$

Tāpiri 196 ki te -132.

$$a=\frac{-\left(-14\right)±\sqrt{64}}{2}$$

Tuhia te pūtakerua o te 64.

$$a=\frac{-\left(-14\right)±8}{2}$$

Ko te tauaro o -14 ko 14.

$$a=\frac{14±8}{2}$$

Nā, me whakaoti te whārite a=\frac{14±8}{2} ina he tāpiri te ±. Tāpiri 14 ki te 8.

$$a=\frac{22}{2}$$

Whakawehe 22 ki te 2.

$$a=11$$

Nā, me whakaoti te whārite a=\frac{14±8}{2} ina he tango te ±. Tango 8 mai i 14.

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

Whakawehe 6 ki te 2.

$$a=3$$

Kua oti te whārite te whakatau.

$$a=11 a=3$$

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.

$$a^{2}-14a+33=0$$

Me tango 33 mai i ngā taha e rua o te whārite.

$$a^{2}-14a+33-33=-33$$

Mā te tango i te 33 i a ia ake anō ka toe ko te 0.

$$a^{2}-14a=-33$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-14a+\left(-7\right)^{2}=-33+\left(-7\right)^{2}$$

Pūrua -7.

$$a^{2}-14a+49=-33+49$$

Tāpiri -33 ki te 49.

$$a^{2}-14a+49=16$$

Tauwehea a^{2}-14a+49. 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}.

$$\left(a-7\right)^{2}=16$$

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

$$\sqrt{\left(a-7\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$a-7=4 a-7=-4$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$a=11 a=3$$