Whakaoti i te x^2+14x+32=-8

$${ x }^{ 2 } +14x+32=-8$$

Whakaoti mō x

$$x=-10$$

$$x=-4$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 8 ki ngā taha e rua.

$$x^{2}+14x+32+8=0$$

Tāpirihia te 32 ki te 8, ka 40.

$$x^{2}+14x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+14x+40 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.

$$a+b=14 ab=40$$

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 40.

$$1,40 2,20 4,10 5,8$$

Tātaihia te tapeke mō ia takirua.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

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

$$a=4 b=10$$

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

$$\left(x+4\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x+4=0 me te x+10=0.

$$x=-4 x=-10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 8 ki ngā taha e rua.

$$x^{2}+14x+32+8=0$$

Tāpirihia te 32 ki te 8, ka 40.

$$x^{2}+14x+40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=14 ab=1\times 40=40$$

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 40.

$$1,40 2,20 4,10 5,8$$

Tātaihia te tapeke mō ia takirua.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

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

$$a=4 b=10$$

Tuhia anō te x^{2}+14x+40 hei \left(x^{2}+4x\right)+\left(10x+40\right).

$$\left(x^{2}+4x\right)+\left(10x+40\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x+4\right)+10\left(x+4\right)$$

Whakatauwehea atu te kīanga pātahi x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+4\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x+4=0 me te x+10=0.

$$x=-4 x=-10$$

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.

$$x^{2}+14x+32=-8$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x^{2}+14x+32-\left(-8\right)=-8-\left(-8\right)$$

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x+32-\left(-8\right)=0$$

Tango -8 mai i 32.

$$x^{2}+14x+40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 14 mō b, me 40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 40}}{2}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 40}}{2}$$

Whakareatia -4 ki te 40.

$$x=\frac{-14±\sqrt{196-160}}{2}$$

Tāpiri 196 ki te -160.

$$x=\frac{-14±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-14±6}{2}$$

Nā, me whakaoti te whārite x=\frac{-14±6}{2} ina he tāpiri te ±. Tāpiri -14 ki te 6.

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

Whakawehe -8 ki te 2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{-14±6}{2} ina he tango te ±. Tango 6 mai i -14.

$$x=-\frac{20}{2}$$

Whakawehe -20 ki te 2.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=-4 x=-10$$

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.

$$x^{2}+14x+32=-8$$

Me tango 32 mai i ngā taha e rua o te whārite.

$$x^{2}+14x+32-32=-8-32$$

Mā te tango i te 32 i a ia ake anō ka toe ko te 0.

$$x^{2}+14x=-8-32$$

Tango 32 mai i -8.

$$x^{2}+14x=-40$$

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.

$$x^{2}+14x+7^{2}=-40+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=-40+49$$

Tāpiri -40 ki te 49.

$$x^{2}+14x+49=9$$

Tauwehea x^{2}+14x+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}.

$$\left(x+7\right)^{2}=9$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+7=3 x+7=-3$$

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

$$x=-4 x=-10$$