Whakaoti i te a^2+10a+21=0

$$a ^ { 2 } + 10 a + 21 = 0$$

Whakaoti mō a

$$a=-7$$

$$a=-3$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te a^{2}+10a+21 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.

$$a+b=10 ab=21$$

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 21.

$$1,21 3,7$$

Tātaihia te tapeke mō ia takirua.

$$1+21=22 3+7=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=3 b=7$$

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

$$\left(a+3\right)\left(a+7\right)$$

Hei kimi otinga whārite, me whakaoti te a+3=0 me te a+7=0.

$$a=-3 a=-7$$

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 a^{2}+aa+ba+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=1\times 21=21$$

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 21.

$$1,21 3,7$$

Tātaihia te tapeke mō ia takirua.

$$1+21=22 3+7=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=3 b=7$$

Tuhia anō te a^{2}+10a+21 hei \left(a^{2}+3a\right)+\left(7a+21\right).

$$\left(a^{2}+3a\right)+\left(7a+21\right)$$

Tauwehea te a i te tuatahi me te 7 i te rōpū tuarua.

$$a\left(a+3\right)+7\left(a+3\right)$$

Whakatauwehea atu te kīanga pātahi a+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a+3\right)\left(a+7\right)$$

Hei kimi otinga whārite, me whakaoti te a+3=0 me te a+7=0.

$$a=-3 a=-7$$

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.

$$a^{2}+10a+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 10 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-10±\sqrt{10^{2}-4\times 21}}{2}$$

Pūrua 10.

$$a=\frac{-10±\sqrt{100-4\times 21}}{2}$$

Whakareatia -4 ki te 21.

$$a=\frac{-10±\sqrt{100-84}}{2}$$

Tāpiri 100 ki te -84.

$$a=\frac{-10±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$a=\frac{-10±4}{2}$$

Nā, me whakaoti te whārite a=\frac{-10±4}{2} ina he tāpiri te ±. Tāpiri -10 ki te 4.

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

Whakawehe -6 ki te 2.

$$a=-3$$

Nā, me whakaoti te whārite a=\frac{-10±4}{2} ina he tango te ±. Tango 4 mai i -10.

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

Whakawehe -14 ki te 2.

$$a=-7$$

Kua oti te whārite te whakatau.

$$a=-3 a=-7$$

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.

$$a^{2}+10a+21=0$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$a^{2}+10a+21-21=-21$$

Mā te tango i te 21 i a ia ake anō ka toe ko te 0.

$$a^{2}+10a=-21$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+10a+5^{2}=-21+5^{2}$$

Pūrua 5.

$$a^{2}+10a+25=-21+25$$

Tāpiri -21 ki te 25.

$$a^{2}+10a+25=4$$

Tauwehea a^{2}+10a+25. 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}.

$$\left(a+5\right)^{2}=4$$

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

$$\sqrt{\left(a+5\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$a+5=2 a+5=-2$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$a=-3 a=-7$$