Whakaoti i te 3x^2=30x+600

$$3 x ^ { 2 } = 30 x + 600$$

Whakaoti mō x

$$x=-10$$

$$x=20$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 30x mai i ngā taha e rua.

$$3x^{2}-30x=600$$

Tangohia te 600 mai i ngā taha e rua.

$$3x^{2}-30x-600=0$$

Whakawehea ngā taha e rua ki te 3.

$$x^{2}-10x-200=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-200. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=1\left(-200\right)=-200$$

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 -200.

$$1,-200 2,-100 4,-50 5,-40 8,-25 10,-20$$

Tātaihia te tapeke mō ia takirua.

$$1-200=-199 2-100=-98 4-50=-46 5-40=-35 8-25=-17 10-20=-10$$

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

$$a=-20 b=10$$

Tuhia anō te x^{2}-10x-200 hei \left(x^{2}-20x\right)+\left(10x-200\right).

$$\left(x^{2}-20x\right)+\left(10x-200\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x-20\right)+10\left(x-20\right)$$

Whakatauwehea atu te kīanga pātahi x-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-20\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-20=0 me te x+10=0.

$$x=20 x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 30x mai i ngā taha e rua.

$$3x^{2}-30x=600$$

Tangohia te 600 mai i ngā taha e rua.

$$3x^{2}-30x-600=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -30 mō b, me -600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 3\left(-600\right)}}{2\times 3}$$

Pūrua -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 3\left(-600\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-30\right)±\sqrt{900-12\left(-600\right)}}{2\times 3}$$

Whakareatia -12 ki te -600.

$$x=\frac{-\left(-30\right)±\sqrt{900+7200}}{2\times 3}$$

Tāpiri 900 ki te 7200.

$$x=\frac{-\left(-30\right)±\sqrt{8100}}{2\times 3}$$

Tuhia te pūtakerua o te 8100.

$$x=\frac{-\left(-30\right)±90}{2\times 3}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±90}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{30±90}{6}$$

Nā, me whakaoti te whārite x=\frac{30±90}{6} ina he tāpiri te ±. Tāpiri 30 ki te 90.

$$x=\frac{120}{6}$$

Whakawehe 120 ki te 6.

$$x=20$$

Nā, me whakaoti te whārite x=\frac{30±90}{6} ina he tango te ±. Tango 90 mai i 30.

$$x=-\frac{60}{6}$$

Whakawehe -60 ki te 6.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=20 x=-10$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 30x mai i ngā taha e rua.

$$3x^{2}-30x=600$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-30x}{3}=\frac{600}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\left(-\frac{30}{3}\right)x=\frac{600}{3}$$

Whakawehe -30 ki te 3.

$$x^{2}-10x=\frac{600}{3}$$

Whakawehe 600 ki te 3.

$$x^{2}-10x=200$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=200+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=200+25$$

Tāpiri 200 ki te 25.

$$x^{2}-10x+25=225$$

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}.

$$\left(x-5\right)^{2}=225$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{225}$$

Whakarūnātia.

$$x-5=15 x-5=-15$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x=20 x=-10$$