Whakaoti i te 2x^2-60x-3600=0

$$2 x ^ { 2 } - 60 x - 3600 = 0$$

Whakaoti mō x

$$x=-30$$

$$x=60$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-30x-1800=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-1800. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-30 ab=1\left(-1800\right)=-1800$$

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 -1800.

$$1,-1800 2,-900 3,-600 4,-450 5,-360 6,-300 8,-225 9,-200 10,-180 12,-150 15,-120 18,-100 20,-90 24,-75 25,-72 30,-60 36,-50 40,-45$$

Tātaihia te tapeke mō ia takirua.

$$1-1800=-1799 2-900=-898 3-600=-597 4-450=-446 5-360=-355 6-300=-294 8-225=-217 9-200=-191 10-180=-170 12-150=-138 15-120=-105 18-100=-82 20-90=-70 24-75=-51 25-72=-47 30-60=-30 36-50=-14 40-45=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -30.

$$a=-60 b=30$$

Tuhia anō te x^{2}-30x-1800 hei \left(x^{2}-60x\right)+\left(30x-1800\right).

$$\left(x^{2}-60x\right)+\left(30x-1800\right)$$

Tauwehea te x i te tuatahi me te 30 i te rōpū tuarua.

$$x\left(x-60\right)+30\left(x-60\right)$$

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

$$\left(x-60\right)\left(x+30\right)$$

Hei kimi otinga whārite, me whakaoti te x-60=0 me te x+30=0.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -60 mō b, me -3600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 2\left(-3600\right)}}{2\times 2}$$

Pūrua -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 2\left(-3600\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-60\right)±\sqrt{3600-8\left(-3600\right)}}{2\times 2}$$

Whakareatia -8 ki te -3600.

$$x=\frac{-\left(-60\right)±\sqrt{3600+28800}}{2\times 2}$$

Tāpiri 3600 ki te 28800.

$$x=\frac{-\left(-60\right)±\sqrt{32400}}{2\times 2}$$

Tuhia te pūtakerua o te 32400.

$$x=\frac{-\left(-60\right)±180}{2\times 2}$$

Ko te tauaro o -60 ko 60.

$$x=\frac{60±180}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{60±180}{4}$$

Nā, me whakaoti te whārite x=\frac{60±180}{4} ina he tāpiri te ±. Tāpiri 60 ki te 180.

$$x=\frac{240}{4}$$

Whakawehe 240 ki te 4.

$$x=60$$

Nā, me whakaoti te whārite x=\frac{60±180}{4} ina he tango te ±. Tango 180 mai i 60.

$$x=-\frac{120}{4}$$

Whakawehe -120 ki te 4.

$$x=-30$$

Kua oti te whārite te whakatau.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

Me tāpiri 3600 ki ngā taha e rua o te whārite.

$$2x^{2}-60x-3600-\left(-3600\right)=-\left(-3600\right)$$

Mā te tango i te -3600 i a ia ake anō ka toe ko te 0.

$$2x^{2}-60x=-\left(-3600\right)$$

Tango -3600 mai i 0.

$$2x^{2}-60x=3600$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-60x}{2}=\frac{3600}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{60}{2}\right)x=\frac{3600}{2}$$

Whakawehe -60 ki te 2.

$$x^{2}-30x=\frac{3600}{2}$$

Whakawehe 3600 ki te 2.

$$x^{2}-30x=1800$$

Whakawehea te -30, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -15. Nā, tāpiria te pūrua o te -15 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-30x+\left(-15\right)^{2}=1800+\left(-15\right)^{2}$$

Pūrua -15.

$$x^{2}-30x+225=1800+225$$

Tāpiri 1800 ki te 225.

$$x^{2}-30x+225=2025$$

Tauwehea x^{2}-30x+225. 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}.

$$\left(x-15\right)^{2}=2025$$

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

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{2025}$$

Whakarūnātia.

$$x-15=45 x-15=-45$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$x=60 x=-30$$