Whakaoti i te x^2+100x-7500=0

$$x ^ { 2 } + 100 x - 7500 = 0$$

Whakaoti mō x

$$x=-150$$

$$x=50$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+100x-7500 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.

$$a+b=100 ab=-7500$$

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 -7500.

$$-1,7500 -2,3750 -3,2500 -4,1875 -5,1500 -6,1250 -10,750 -12,625 -15,500 -20,375 -25,300 -30,250 -50,150 -60,125 -75,100$$

Tātaihia te tapeke mō ia takirua.

$$-1+7500=7499 -2+3750=3748 -3+2500=2497 -4+1875=1871 -5+1500=1495 -6+1250=1244 -10+750=740 -12+625=613 -15+500=485 -20+375=355 -25+300=275 -30+250=220 -50+150=100 -60+125=65 -75+100=25$$

Ko te otinga te takirua ka hoatu i te tapeke 100.

$$a=-50 b=150$$

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

$$\left(x-50\right)\left(x+150\right)$$

Hei kimi otinga whārite, me whakaoti te x-50=0 me te x+150=0.

$$x=50 x=-150$$

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-7500. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=100 ab=1\left(-7500\right)=-7500$$

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 -7500.

$$-1,7500 -2,3750 -3,2500 -4,1875 -5,1500 -6,1250 -10,750 -12,625 -15,500 -20,375 -25,300 -30,250 -50,150 -60,125 -75,100$$

Tātaihia te tapeke mō ia takirua.

$$-1+7500=7499 -2+3750=3748 -3+2500=2497 -4+1875=1871 -5+1500=1495 -6+1250=1244 -10+750=740 -12+625=613 -15+500=485 -20+375=355 -25+300=275 -30+250=220 -50+150=100 -60+125=65 -75+100=25$$

Ko te otinga te takirua ka hoatu i te tapeke 100.

$$a=-50 b=150$$

Tuhia anō te x^{2}+100x-7500 hei \left(x^{2}-50x\right)+\left(150x-7500\right).

$$\left(x^{2}-50x\right)+\left(150x-7500\right)$$

Tauwehea te x i te tuatahi me te 150 i te rōpū tuarua.

$$x\left(x-50\right)+150\left(x-50\right)$$

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

$$\left(x-50\right)\left(x+150\right)$$

Hei kimi otinga whārite, me whakaoti te x-50=0 me te x+150=0.

$$x=50 x=-150$$

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.

$$x^{2}+100x-7500=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 100 mō b, me -7500 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-7500\right)}}{2}$$

Pūrua 100.

$$x=\frac{-100±\sqrt{10000-4\left(-7500\right)}}{2}$$

Whakareatia -4 ki te -7500.

$$x=\frac{-100±\sqrt{10000+30000}}{2}$$

Tāpiri 10000 ki te 30000.

$$x=\frac{-100±\sqrt{40000}}{2}$$

Tuhia te pūtakerua o te 40000.

$$x=\frac{-100±200}{2}$$

Nā, me whakaoti te whārite x=\frac{-100±200}{2} ina he tāpiri te ±. Tāpiri -100 ki te 200.

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

Whakawehe 100 ki te 2.

$$x=50$$

Nā, me whakaoti te whārite x=\frac{-100±200}{2} ina he tango te ±. Tango 200 mai i -100.

$$x=-\frac{300}{2}$$

Whakawehe -300 ki te 2.

$$x=-150$$

Kua oti te whārite te whakatau.

$$x=50 x=-150$$

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.

$$x^{2}+100x-7500=0$$

Me tāpiri 7500 ki ngā taha e rua o te whārite.

$$x^{2}+100x-7500-\left(-7500\right)=-\left(-7500\right)$$

Mā te tango i te -7500 i a ia ake anō ka toe ko te 0.

$$x^{2}+100x=-\left(-7500\right)$$

Tango -7500 mai i 0.

$$x^{2}+100x=7500$$

Whakawehea te 100, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 50. Nā, tāpiria te pūrua o te 50 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+100x+50^{2}=7500+50^{2}$$

Pūrua 50.

$$x^{2}+100x+2500=7500+2500$$

Tāpiri 7500 ki te 2500.

$$x^{2}+100x+2500=10000$$

Tauwehea x^{2}+100x+2500. 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}.

$$\left(x+50\right)^{2}=10000$$

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

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{10000}$$

Whakarūnātia.

$$x+50=100 x+50=-100$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$x=50 x=-150$$