Whakaoti i te x^2-50x-5000=0

$$x ^ { 2 } - 50 x - 5000 = 0$$

Whakaoti mō x

$$x=-50$$

$$x=100$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-50x-5000 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.

$$a+b=-50 ab=-5000$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -5000.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Tātaihia te tapeke mō ia takirua.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

Ko te otinga te takirua ka hoatu i te tapeke -50.

$$a=-100 b=50$$

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

$$\left(x-100\right)\left(x+50\right)$$

Hei kimi otinga whārite, me whakaoti te x-100=0 me te x+50=0.

$$x=100 x=-50$$

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-5000. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-50 ab=1\left(-5000\right)=-5000$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -5000.

$$1,-5000 2,-2500 4,-1250 5,-1000 8,-625 10,-500 20,-250 25,-200 40,-125 50,-100$$

Tātaihia te tapeke mō ia takirua.

$$1-5000=-4999 2-2500=-2498 4-1250=-1246 5-1000=-995 8-625=-617 10-500=-490 20-250=-230 25-200=-175 40-125=-85 50-100=-50$$

Ko te otinga te takirua ka hoatu i te tapeke -50.

$$a=-100 b=50$$

Tuhia anō te x^{2}-50x-5000 hei \left(x^{2}-100x\right)+\left(50x-5000\right).

$$\left(x^{2}-100x\right)+\left(50x-5000\right)$$

Tauwehea te x i te tuatahi me te 50 i te rōpū tuarua.

$$x\left(x-100\right)+50\left(x-100\right)$$

Whakatauwehea atu te kīanga pātahi x-100 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-100\right)\left(x+50\right)$$

Hei kimi otinga whārite, me whakaoti te x-100=0 me te x+50=0.

$$x=100 x=-50$$

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.

$$x^{2}-50x-5000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -50 mō b, me -5000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-5000\right)}}{2}$$

Pūrua -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-5000\right)}}{2}$$

Whakareatia -4 ki te -5000.

$$x=\frac{-\left(-50\right)±\sqrt{2500+20000}}{2}$$

Tāpiri 2500 ki te 20000.

$$x=\frac{-\left(-50\right)±\sqrt{22500}}{2}$$

Tuhia te pūtakerua o te 22500.

$$x=\frac{-\left(-50\right)±150}{2}$$

Ko te tauaro o -50 ko 50.

$$x=\frac{50±150}{2}$$

Nā, me whakaoti te whārite x=\frac{50±150}{2} ina he tāpiri te ±. Tāpiri 50 ki te 150.

$$x=\frac{200}{2}$$

Whakawehe 200 ki te 2.

$$x=100$$

Nā, me whakaoti te whārite x=\frac{50±150}{2} ina he tango te ±. Tango 150 mai i 50.

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

Whakawehe -100 ki te 2.

$$x=-50$$

Kua oti te whārite te whakatau.

$$x=100 x=-50$$

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.

$$x^{2}-50x-5000=0$$

Me tāpiri 5000 ki ngā taha e rua o te whārite.

$$x^{2}-50x-5000-\left(-5000\right)=-\left(-5000\right)$$

Mā te tango i te -5000 i a ia ake anō ka toe ko te 0.

$$x^{2}-50x=-\left(-5000\right)$$

Tango -5000 mai i 0.

$$x^{2}-50x=5000$$

Whakawehea te -50, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -25. Nā, tāpiria te pūrua o te -25 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-50x+\left(-25\right)^{2}=5000+\left(-25\right)^{2}$$

Pūrua -25.

$$x^{2}-50x+625=5000+625$$

Tāpiri 5000 ki te 625.

$$x^{2}-50x+625=5625$$

Tauwehea x^{2}-50x+625. 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}.

$$\left(x-25\right)^{2}=5625$$

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

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{5625}$$

Whakarūnātia.

$$x-25=75 x-25=-75$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$x=100 x=-50$$