Whakaoti i te x^2+10x-3000=0

$$x ^ { 2 } + 10 x - 3000 = 0$$

Whakaoti mō x

$$x=-60$$

$$x=50$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+10x-3000 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.

$$a+b=10 ab=-3000$$

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 -3000.

$$-1,3000 -2,1500 -3,1000 -4,750 -5,600 -6,500 -8,375 -10,300 -12,250 -15,200 -20,150 -24,125 -25,120 -30,100 -40,75 -50,60$$

Tātaihia te tapeke mō ia takirua.

$$-1+3000=2999 -2+1500=1498 -3+1000=997 -4+750=746 -5+600=595 -6+500=494 -8+375=367 -10+300=290 -12+250=238 -15+200=185 -20+150=130 -24+125=101 -25+120=95 -30+100=70 -40+75=35 -50+60=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-50 b=60$$

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

$$\left(x-50\right)\left(x+60\right)$$

Hei kimi otinga whārite, me whakaoti te x-50=0 me te x+60=0.

$$x=50 x=-60$$

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-3000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=1\left(-3000\right)=-3000$$

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 -3000.

$$-1,3000 -2,1500 -3,1000 -4,750 -5,600 -6,500 -8,375 -10,300 -12,250 -15,200 -20,150 -24,125 -25,120 -30,100 -40,75 -50,60$$

Tātaihia te tapeke mō ia takirua.

$$-1+3000=2999 -2+1500=1498 -3+1000=997 -4+750=746 -5+600=595 -6+500=494 -8+375=367 -10+300=290 -12+250=238 -15+200=185 -20+150=130 -24+125=101 -25+120=95 -30+100=70 -40+75=35 -50+60=10$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-50 b=60$$

Tuhia anō te x^{2}+10x-3000 hei \left(x^{2}-50x\right)+\left(60x-3000\right).

$$\left(x^{2}-50x\right)+\left(60x-3000\right)$$

Tauwehea te x i te tuatahi me te 60 i te rōpū tuarua.

$$x\left(x-50\right)+60\left(x-50\right)$$

Whakatauwehea atu te kīanga pātahi x-50 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-50\right)\left(x+60\right)$$

Hei kimi otinga whārite, me whakaoti te x-50=0 me te x+60=0.

$$x=50 x=-60$$

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.

$$x^{2}+10x-3000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 10 mō b, me -3000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-3000\right)}}{2}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\left(-3000\right)}}{2}$$

Whakareatia -4 ki te -3000.

$$x=\frac{-10±\sqrt{100+12000}}{2}$$

Tāpiri 100 ki te 12000.

$$x=\frac{-10±\sqrt{12100}}{2}$$

Tuhia te pūtakerua o te 12100.

$$x=\frac{-10±110}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±110}{2} ina he tāpiri te ±. Tāpiri -10 ki te 110.

$$x=\frac{100}{2}$$

Whakawehe 100 ki te 2.

$$x=50$$

Nā, me whakaoti te whārite x=\frac{-10±110}{2} ina he tango te ±. Tango 110 mai i -10.

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

Whakawehe -120 ki te 2.

$$x=-60$$

Kua oti te whārite te whakatau.

$$x=50 x=-60$$

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.

$$x^{2}+10x-3000=0$$

Me tāpiri 3000 ki ngā taha e rua o te whārite.

$$x^{2}+10x-3000-\left(-3000\right)=-\left(-3000\right)$$

Mā te tango i te -3000 i a ia ake anō ka toe ko te 0.

$$x^{2}+10x=-\left(-3000\right)$$

Tango -3000 mai i 0.

$$x^{2}+10x=3000$$

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.

$$x^{2}+10x+5^{2}=3000+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=3000+25$$

Tāpiri 3000 ki te 25.

$$x^{2}+10x+25=3025$$

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}.

$$\left(x+5\right)^{2}=3025$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{3025}$$

Whakarūnātia.

$$x+5=55 x+5=-55$$

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

$$x=50 x=-60$$