Whakaoti i te x^2+8x+37=21

$${ x }^{ 2 } +8x+37=21$$

Whakaoti mō x

$$x=-4$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 21 mai i ngā taha e rua.

$$x^{2}+8x+37-21=0$$

Tangohia te 21 i te 37, ka 16.

$$x^{2}+8x+16=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+8x+16 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.

$$a+b=8 ab=16$$

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 16.

$$1,16 2,8 4,4$$

Tātaihia te tapeke mō ia takirua.

$$1+16=17 2+8=10 4+4=8$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=4 b=4$$

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

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x+4=0.

$$x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 21 mai i ngā taha e rua.

$$x^{2}+8x+37-21=0$$

Tangohia te 21 i te 37, ka 16.

$$x^{2}+8x+16=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=8 ab=1\times 16=16$$

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 16.

$$1,16 2,8 4,4$$

Tātaihia te tapeke mō ia takirua.

$$1+16=17 2+8=10 4+4=8$$

Ko te otinga te takirua ka hoatu i te tapeke 8.

$$a=4 b=4$$

Tuhia anō te x^{2}+8x+16 hei \left(x^{2}+4x\right)+\left(4x+16\right).

$$\left(x^{2}+4x\right)+\left(4x+16\right)$$

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x+4=0.

$$x=-4$$

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.

$$x^{2}+8x+37=21$$

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

$$x^{2}+8x+37-21=21-21$$

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

$$x^{2}+8x+37-21=0$$

Tango 21 mai i 37.

$$x^{2}+8x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 8 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 16}}{2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 16}}{2}$$

Whakareatia -4 ki te 16.

$$x=\frac{-8±\sqrt{64-64}}{2}$$

Tāpiri 64 ki te -64.

$$x=\frac{-8±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

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

Whakawehe -8 ki te 2.

$$x=-4$$

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.

$$x^{2}+8x+37=21$$

Me tango 37 mai i ngā taha e rua o te whārite.

$$x^{2}+8x+37-37=21-37$$

Mā te tango i te 37 i a ia ake anō ka toe ko te 0.

$$x^{2}+8x=21-37$$

Tango 37 mai i 21.

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

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.

$$x^{2}+8x+4^{2}=-16+4^{2}$$

Pūrua 4.

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

Tāpiri -16 ki te 16.

$$x^{2}+8x+16=0$$

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}.

$$\left(x+4\right)^{2}=0$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+4=0 x+4=0$$

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

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

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=-4$$