Whakaoti i te x^2+16x-512=0

$$x ^ { 2 } + 16 x - 512 = 0$$

Whakaoti mō x

$$x=-32$$

$$x=16$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+16x-512 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.

$$a+b=16 ab=-512$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -512.

$$-1,512 -2,256 -4,128 -8,64 -16,32$$

Tātaihia te tapeke mō ia takirua.

$$-1+512=511 -2+256=254 -4+128=124 -8+64=56 -16+32=16$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=-16 b=32$$

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

$$\left(x-16\right)\left(x+32\right)$$

Hei kimi otinga whārite, me whakaoti te x-16=0 me te x+32=0.

$$x=16 x=-32$$

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-512. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=16 ab=1\left(-512\right)=-512$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -512.

$$-1,512 -2,256 -4,128 -8,64 -16,32$$

Tātaihia te tapeke mō ia takirua.

$$-1+512=511 -2+256=254 -4+128=124 -8+64=56 -16+32=16$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=-16 b=32$$

Tuhia anō te x^{2}+16x-512 hei \left(x^{2}-16x\right)+\left(32x-512\right).

$$\left(x^{2}-16x\right)+\left(32x-512\right)$$

Tauwehea te x i te tuatahi me te 32 i te rōpū tuarua.

$$x\left(x-16\right)+32\left(x-16\right)$$

Whakatauwehea atu te kīanga pātahi x-16 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-16\right)\left(x+32\right)$$

Hei kimi otinga whārite, me whakaoti te x-16=0 me te x+32=0.

$$x=16 x=-32$$

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.

$$x^{2}+16x-512=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 16 mō b, me -512 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-512\right)}}{2}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\left(-512\right)}}{2}$$

Whakareatia -4 ki te -512.

$$x=\frac{-16±\sqrt{256+2048}}{2}$$

Tāpiri 256 ki te 2048.

$$x=\frac{-16±\sqrt{2304}}{2}$$

Tuhia te pūtakerua o te 2304.

$$x=\frac{-16±48}{2}$$

Nā, me whakaoti te whārite x=\frac{-16±48}{2} ina he tāpiri te ±. Tāpiri -16 ki te 48.

$$x=\frac{32}{2}$$

Whakawehe 32 ki te 2.

$$x=16$$

Nā, me whakaoti te whārite x=\frac{-16±48}{2} ina he tango te ±. Tango 48 mai i -16.

$$x=-\frac{64}{2}$$

Whakawehe -64 ki te 2.

$$x=-32$$

Kua oti te whārite te whakatau.

$$x=16 x=-32$$

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.

$$x^{2}+16x-512=0$$

Me tāpiri 512 ki ngā taha e rua o te whārite.

$$x^{2}+16x-512-\left(-512\right)=-\left(-512\right)$$

Mā te tango i te -512 i a ia ake anō ka toe ko te 0.

$$x^{2}+16x=-\left(-512\right)$$

Tango -512 mai i 0.

$$x^{2}+16x=512$$

Whakawehea te 16, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 8. Nā, tāpiria te pūrua o te 8 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+16x+8^{2}=512+8^{2}$$

Pūrua 8.

$$x^{2}+16x+64=512+64$$

Tāpiri 512 ki te 64.

$$x^{2}+16x+64=576$$

Tauwehea x^{2}+16x+64. 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}.

$$\left(x+8\right)^{2}=576$$

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

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{576}$$

Whakarūnātia.

$$x+8=24 x+8=-24$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$x=16 x=-32$$