Whakaoti i te x^2+34x-71000=0

$$x ^ { 2 } + 34 x - 71000 = 0$$

Whakaoti mō x

$$x=-284$$

$$x=250$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+34x-71000 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.

$$a+b=34 ab=-71000$$

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 -71000.

$$-1,71000 -2,35500 -4,17750 -5,14200 -8,8875 -10,7100 -20,3550 -25,2840 -40,1775 -50,1420 -71,1000 -100,710 -125,568 -142,500 -200,355 -250,284$$

Tātaihia te tapeke mō ia takirua.

$$-1+71000=70999 -2+35500=35498 -4+17750=17746 -5+14200=14195 -8+8875=8867 -10+7100=7090 -20+3550=3530 -25+2840=2815 -40+1775=1735 -50+1420=1370 -71+1000=929 -100+710=610 -125+568=443 -142+500=358 -200+355=155 -250+284=34$$

Ko te otinga te takirua ka hoatu i te tapeke 34.

$$a=-250 b=284$$

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

$$\left(x-250\right)\left(x+284\right)$$

Hei kimi otinga whārite, me whakaoti te x-250=0 me te x+284=0.

$$x=250 x=-284$$

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-71000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=34 ab=1\left(-71000\right)=-71000$$

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 -71000.

$$-1,71000 -2,35500 -4,17750 -5,14200 -8,8875 -10,7100 -20,3550 -25,2840 -40,1775 -50,1420 -71,1000 -100,710 -125,568 -142,500 -200,355 -250,284$$

Tātaihia te tapeke mō ia takirua.

$$-1+71000=70999 -2+35500=35498 -4+17750=17746 -5+14200=14195 -8+8875=8867 -10+7100=7090 -20+3550=3530 -25+2840=2815 -40+1775=1735 -50+1420=1370 -71+1000=929 -100+710=610 -125+568=443 -142+500=358 -200+355=155 -250+284=34$$

Ko te otinga te takirua ka hoatu i te tapeke 34.

$$a=-250 b=284$$

Tuhia anō te x^{2}+34x-71000 hei \left(x^{2}-250x\right)+\left(284x-71000\right).

$$\left(x^{2}-250x\right)+\left(284x-71000\right)$$

Tauwehea te x i te tuatahi me te 284 i te rōpū tuarua.

$$x\left(x-250\right)+284\left(x-250\right)$$

Whakatauwehea atu te kīanga pātahi x-250 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-250\right)\left(x+284\right)$$

Hei kimi otinga whārite, me whakaoti te x-250=0 me te x+284=0.

$$x=250 x=-284$$

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.

$$x^{2}+34x-71000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 34 mō b, me -71000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-71000\right)}}{2}$$

Pūrua 34.

$$x=\frac{-34±\sqrt{1156-4\left(-71000\right)}}{2}$$

Whakareatia -4 ki te -71000.

$$x=\frac{-34±\sqrt{1156+284000}}{2}$$

Tāpiri 1156 ki te 284000.

$$x=\frac{-34±\sqrt{285156}}{2}$$

Tuhia te pūtakerua o te 285156.

$$x=\frac{-34±534}{2}$$

Nā, me whakaoti te whārite x=\frac{-34±534}{2} ina he tāpiri te ±. Tāpiri -34 ki te 534.

$$x=\frac{500}{2}$$

Whakawehe 500 ki te 2.

$$x=250$$

Nā, me whakaoti te whārite x=\frac{-34±534}{2} ina he tango te ±. Tango 534 mai i -34.

$$x=-\frac{568}{2}$$

Whakawehe -568 ki te 2.

$$x=-284$$

Kua oti te whārite te whakatau.

$$x=250 x=-284$$

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.

$$x^{2}+34x-71000=0$$

Me tāpiri 71000 ki ngā taha e rua o te whārite.

$$x^{2}+34x-71000-\left(-71000\right)=-\left(-71000\right)$$

Mā te tango i te -71000 i a ia ake anō ka toe ko te 0.

$$x^{2}+34x=-\left(-71000\right)$$

Tango -71000 mai i 0.

$$x^{2}+34x=71000$$

Whakawehea te 34, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 17. Nā, tāpiria te pūrua o te 17 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+34x+17^{2}=71000+17^{2}$$

Pūrua 17.

$$x^{2}+34x+289=71000+289$$

Tāpiri 71000 ki te 289.

$$x^{2}+34x+289=71289$$

Tauwehea x^{2}+34x+289. 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}.

$$\left(x+17\right)^{2}=71289$$

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

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{71289}$$

Whakarūnātia.

$$x+17=267 x+17=-267$$

Me tango 17 mai i ngā taha e rua o te whārite.

$$x=250 x=-284$$