Whakaoti i te x^2+8x+7=0

$$x ^ { 2 } + 8 x + 7 = 0$$

Whakaoti mō x

$$x=-7$$

$$x=-1$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+8x+7 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.

$$a+b=8 ab=7$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=7$$

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

$$\left(x+1\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x+1=0 me te x+7=0.

$$x=-1 x=-7$$

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 x^{2}+ax+bx+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=1\times 7=7$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=7$$

Tuhia anō te x^{2}+8x+7 hei \left(x^{2}+x\right)+\left(7x+7\right).

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

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

$$x\left(x+1\right)+7\left(x+1\right)$$

Whakatauwehea atu te kīanga pātahi x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+1\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x+1=0 me te x+7=0.

$$x=-1 x=-7$$

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.

$$x^{2}+8x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 8 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 7}}{2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 7}}{2}$$

Whakareatia -4 ki te 7.

$$x=\frac{-8±\sqrt{64-28}}{2}$$

Tāpiri 64 ki te -28.

$$x=\frac{-8±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-8±6}{2}$$

Nā, me whakaoti te whārite x=\frac{-8±6}{2} ina he tāpiri te ±. Tāpiri -8 ki te 6.

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

Whakawehe -2 ki te 2.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-8±6}{2} ina he tango te ±. Tango 6 mai i -8.

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

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=-1 x=-7$$

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.

$$x^{2}+8x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x^{2}+8x+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$x^{2}+8x=-7$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=-7+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=-7+16$$

Tāpiri -7 ki te 16.

$$x^{2}+8x+16=9$$

Tauwehea x^{2}+8x+16. 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}.

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

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

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

Whakarūnātia.

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

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=-1 x=-7$$