Whakaoti i te x^2+7x+12=0

$$x ^ { 2 } + 7 x + 12 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=-3$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+7x+12 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.

$$a+b=7 ab=12$$

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 12.

$$1,12 2,6 3,4$$

Tātaihia te tapeke mō ia takirua.

$$1+12=13 2+6=8 3+4=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=3 b=4$$

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

$$\left(x+3\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x+3=0 me te x+4=0.

$$x=-3 x=-4$$

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+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=1\times 12=12$$

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 12.

$$1,12 2,6 3,4$$

Tātaihia te tapeke mō ia takirua.

$$1+12=13 2+6=8 3+4=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=3 b=4$$

Tuhia anō te x^{2}+7x+12 hei \left(x^{2}+3x\right)+\left(4x+12\right).

$$\left(x^{2}+3x\right)+\left(4x+12\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x+3\right)+4\left(x+3\right)$$

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

$$\left(x+3\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x+3=0 me te x+4=0.

$$x=-3 x=-4$$

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.

$$x^{2}+7x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 7 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 12}}{2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$x=\frac{-7±\sqrt{49-48}}{2}$$

Tāpiri 49 ki te -48.

$$x=\frac{-7±\sqrt{1}}{2}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-7±1}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±1}{2} ina he tāpiri te ±. Tāpiri -7 ki te 1.

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

Whakawehe -6 ki te 2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-7±1}{2} ina he tango te ±. Tango 1 mai i -7.

$$x=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=-3 x=-4$$

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.

$$x^{2}+7x+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$x^{2}+7x+12-12=-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$x^{2}+7x=-12$$

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

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-12+\left(\frac{7}{2}\right)^{2}$$

Pūruatia \frac{7}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+7x+\frac{49}{4}=-12+\frac{49}{4}$$

Tāpiri -12 ki te \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{1}{4}$$

Tauwehea te x^{2}+7x+\frac{49}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$x+\frac{7}{2}=\frac{1}{2} x+\frac{7}{2}=-\frac{1}{2}$$

Me tango \frac{7}{2} mai i ngā taha e rua o te whārite.

$$x=-3 x=-4$$