Whakaoti i te x^2-6x-16=0

$${ x }^{ 2 } -6x-16 = 0$$

Whakaoti mō x

$$x=-2$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x-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.

$$a+b=-6 ab=-16$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-16=-15 2-8=-6 4-4=0$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-8 b=2$$

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

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

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

$$x=8 x=-2$$

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-16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\left(-16\right)=-16$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-16=-15 2-8=-6 4-4=0$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-8 b=2$$

Tuhia anō te x^{2}-6x-16 hei \left(x^{2}-8x\right)+\left(2x-16\right).

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

Tauwehea te x i te tuatahi me te 2 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=8 x=-2$$

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.

$$x^{2}-6x-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-16\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-16\right)}}{2}$$

Whakareatia -4 ki te -16.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2}$$

Tāpiri 36 ki te 64.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-6\right)±10}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±10}{2}$$

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

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

Whakawehe 16 ki te 2.

$$x=8$$

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

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

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=8 x=-2$$

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.

$$x^{2}-6x-16=0$$

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

$$x^{2}-6x-16-\left(-16\right)=-\left(-16\right)$$

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

$$x^{2}-6x=-\left(-16\right)$$

Tango -16 mai i 0.

$$x^{2}-6x=16$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=16+9$$

Tāpiri 16 ki te 9.

$$x^{2}-6x+9=25$$

Tauwehea te x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=25$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

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

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=8 x=-2$$