Whakaoti i te x^2+2x-3720=0

$$x ^ { 2 } + 2 x - 3720 = 0$$

Whakaoti mō x

$$x=-62$$

$$x=60$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+2x-3720 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.

$$a+b=2 ab=-3720$$

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 -3720.

$$-1,3720 -2,1860 -3,1240 -4,930 -5,744 -6,620 -8,465 -10,372 -12,310 -15,248 -20,186 -24,155 -30,124 -31,120 -40,93 -60,62$$

Tātaihia te tapeke mō ia takirua.

$$-1+3720=3719 -2+1860=1858 -3+1240=1237 -4+930=926 -5+744=739 -6+620=614 -8+465=457 -10+372=362 -12+310=298 -15+248=233 -20+186=166 -24+155=131 -30+124=94 -31+120=89 -40+93=53 -60+62=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-60 b=62$$

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

$$\left(x-60\right)\left(x+62\right)$$

Hei kimi otinga whārite, me whakaoti te x-60=0 me te x+62=0.

$$x=60 x=-62$$

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-3720. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=1\left(-3720\right)=-3720$$

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 -3720.

$$-1,3720 -2,1860 -3,1240 -4,930 -5,744 -6,620 -8,465 -10,372 -12,310 -15,248 -20,186 -24,155 -30,124 -31,120 -40,93 -60,62$$

Tātaihia te tapeke mō ia takirua.

$$-1+3720=3719 -2+1860=1858 -3+1240=1237 -4+930=926 -5+744=739 -6+620=614 -8+465=457 -10+372=362 -12+310=298 -15+248=233 -20+186=166 -24+155=131 -30+124=94 -31+120=89 -40+93=53 -60+62=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-60 b=62$$

Tuhia anō te x^{2}+2x-3720 hei \left(x^{2}-60x\right)+\left(62x-3720\right).

$$\left(x^{2}-60x\right)+\left(62x-3720\right)$$

Tauwehea te x i te tuatahi me te 62 i te rōpū tuarua.

$$x\left(x-60\right)+62\left(x-60\right)$$

Whakatauwehea atu te kīanga pātahi x-60 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-60\right)\left(x+62\right)$$

Hei kimi otinga whārite, me whakaoti te x-60=0 me te x+62=0.

$$x=60 x=-62$$

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.

$$x^{2}+2x-3720=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 2 mō b, me -3720 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3720\right)}}{2}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-3720\right)}}{2}$$

Whakareatia -4 ki te -3720.

$$x=\frac{-2±\sqrt{4+14880}}{2}$$

Tāpiri 4 ki te 14880.

$$x=\frac{-2±\sqrt{14884}}{2}$$

Tuhia te pūtakerua o te 14884.

$$x=\frac{-2±122}{2}$$

Nā, me whakaoti te whārite x=\frac{-2±122}{2} ina he tāpiri te ±. Tāpiri -2 ki te 122.

$$x=\frac{120}{2}$$

Whakawehe 120 ki te 2.

$$x=60$$

Nā, me whakaoti te whārite x=\frac{-2±122}{2} ina he tango te ±. Tango 122 mai i -2.

$$x=-\frac{124}{2}$$

Whakawehe -124 ki te 2.

$$x=-62$$

Kua oti te whārite te whakatau.

$$x=60 x=-62$$

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.

$$x^{2}+2x-3720=0$$

Me tāpiri 3720 ki ngā taha e rua o te whārite.

$$x^{2}+2x-3720-\left(-3720\right)=-\left(-3720\right)$$

Mā te tango i te -3720 i a ia ake anō ka toe ko te 0.

$$x^{2}+2x=-\left(-3720\right)$$

Tango -3720 mai i 0.

$$x^{2}+2x=3720$$

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=3720+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=3720+1$$

Tāpiri 3720 ki te 1.

$$x^{2}+2x+1=3721$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=3721$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{3721}$$

Whakarūnātia.

$$x+1=61 x+1=-61$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=60 x=-62$$