Whakaoti i te 6x^2-12x-210=0

$$6 x ^ { 2 } - 12 x - 210 = 0$$

Whakaoti mō x

$$x=-5$$

$$x=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 6.

$$x^{2}-2x-35=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-35\right)=-35$$

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 -35.

$$1,-35 5,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-35=-34 5-7=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-7 b=5$$

Tuhia anō te x^{2}-2x-35 hei \left(x^{2}-7x\right)+\left(5x-35\right).

$$\left(x^{2}-7x\right)+\left(5x-35\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(x-7\right)+5\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te x+5=0.

$$x=7 x=-5$$

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.

$$6x^{2}-12x-210=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -12 mō b, me -210 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-210\right)}}{2\times 6}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-210\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-24\left(-210\right)}}{2\times 6}$$

Whakareatia -24 ki te -210.

$$x=\frac{-\left(-12\right)±\sqrt{144+5040}}{2\times 6}$$

Tāpiri 144 ki te 5040.

$$x=\frac{-\left(-12\right)±\sqrt{5184}}{2\times 6}$$

Tuhia te pūtakerua o te 5184.

$$x=\frac{-\left(-12\right)±72}{2\times 6}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±72}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{12±72}{12}$$

Nā, me whakaoti te whārite x=\frac{12±72}{12} ina he tāpiri te ±. Tāpiri 12 ki te 72.

$$x=\frac{84}{12}$$

Whakawehe 84 ki te 12.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{12±72}{12} ina he tango te ±. Tango 72 mai i 12.

$$x=-\frac{60}{12}$$

Whakawehe -60 ki te 12.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=7 x=-5$$

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.

$$6x^{2}-12x-210=0$$

Me tāpiri 210 ki ngā taha e rua o te whārite.

$$6x^{2}-12x-210-\left(-210\right)=-\left(-210\right)$$

Mā te tango i te -210 i a ia ake anō ka toe ko te 0.

$$6x^{2}-12x=-\left(-210\right)$$

Tango -210 mai i 0.

$$6x^{2}-12x=210$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-12x}{6}=\frac{210}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\left(-\frac{12}{6}\right)x=\frac{210}{6}$$

Whakawehe -12 ki te 6.

$$x^{2}-2x=\frac{210}{6}$$

Whakawehe 210 ki te 6.

$$x^{2}-2x=35$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=35+1$$

Tāpiri 35 ki te 1.

$$x^{2}-2x+1=36$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=36$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-1=6 x-1=-6$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=7 x=-5$$