Whakaoti i te x^2+10x=-16

$${ x }^{ 2 } +10x=-16$$

Whakaoti mō x

$$x=-8$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 16 ki ngā taha e rua.

$$x^{2}+10x+16=0$$

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

$$a+b=10 ab=16$$

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 10.

$$a=2 b=8$$

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

$$\left(x+2\right)\left(x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+8=0.

$$x=-2 x=-8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 16 ki ngā taha e rua.

$$x^{2}+10x+16=0$$

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.

$$a+b=10 ab=1\times 16=16$$

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 10.

$$a=2 b=8$$

Tuhia anō te x^{2}+10x+16 hei \left(x^{2}+2x\right)+\left(8x+16\right).

$$\left(x^{2}+2x\right)+\left(8x+16\right)$$

Tauwehea te x i te tuatahi me te 8 i te rōpū tuarua.

$$x\left(x+2\right)+8\left(x+2\right)$$

Whakatauwehea atu te kīanga pātahi x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+2\right)\left(x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x+2=0 me te x+8=0.

$$x=-2 x=-8$$

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.

$$x^{2}+10x=-16$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$x^{2}+10x-\left(-16\right)=-16-\left(-16\right)$$

Mā te tango i te -16 i a ia ake anō ka toe ko te 0.

$$x^{2}+10x-\left(-16\right)=0$$

Tango -16 mai i 0.

$$x^{2}+10x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 10 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 16}}{2}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 16}}{2}$$

Whakareatia -4 ki te 16.

$$x=\frac{-10±\sqrt{100-64}}{2}$$

Tāpiri 100 ki te -64.

$$x=\frac{-10±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-10±6}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±6}{2} ina he tāpiri te ±. Tāpiri -10 ki te 6.

$$x=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-10±6}{2} ina he tango te ±. Tango 6 mai i -10.

$$x=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$x=-8$$

Kua oti te whārite te whakatau.

$$x=-2 x=-8$$

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.

$$x^{2}+10x=-16$$

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.

$$x^{2}+10x+5^{2}=-16+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=-16+25$$

Tāpiri -16 ki te 25.

$$x^{2}+10x+25=9$$

Tauwehea x^{2}+10x+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}.

$$\left(x+5\right)^{2}=9$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+5=3 x+5=-3$$

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

$$x=-2 x=-8$$