Whakaoti i te 2x^2-4x-30=0

$$2 x ^ { 2 } - 4 x - 30 = 0$$

Whakaoti mō x

$$x=-3$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-2x-15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-15\right)=-15$$

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 -15.

$$1,-15 3,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-15=-14 3-5=-2$$

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

$$a=-5 b=3$$

Tuhia anō te x^{2}-2x-15 hei \left(x^{2}-5x\right)+\left(3x-15\right).

$$\left(x^{2}-5x\right)+\left(3x-15\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-5\right)+3\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+3=0.

$$x=5 x=-3$$

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.

$$2x^{2}-4x-30=0$$

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

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

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-30\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-30\right)}}{2\times 2}$$

Whakareatia -8 ki te -30.

$$x=\frac{-\left(-4\right)±\sqrt{16+240}}{2\times 2}$$

Tāpiri 16 ki te 240.

$$x=\frac{-\left(-4\right)±\sqrt{256}}{2\times 2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-4\right)±16}{2\times 2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±16}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{4±16}{4}$$

Nā, me whakaoti te whārite x=\frac{4±16}{4} ina he tāpiri te ±. Tāpiri 4 ki te 16.

$$x=\frac{20}{4}$$

Whakawehe 20 ki te 4.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{4±16}{4} ina he tango te ±. Tango 16 mai i 4.

$$x=-\frac{12}{4}$$

Whakawehe -12 ki te 4.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=5 x=-3$$

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.

$$2x^{2}-4x-30=0$$

Me tāpiri 30 ki ngā taha e rua o te whārite.

$$2x^{2}-4x-30-\left(-30\right)=-\left(-30\right)$$

Mā te tango i te -30 i a ia ake anō ka toe ko te 0.

$$2x^{2}-4x=-\left(-30\right)$$

Tango -30 mai i 0.

$$2x^{2}-4x=30$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-4x}{2}=\frac{30}{2}$$

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

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{30}{2}$$

Whakawehe -4 ki te 2.

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

Whakawehe 30 ki te 2.

$$x^{2}-2x=15$$

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.

$$x^{2}-2x+1=15+1$$

Tāpiri 15 ki te 1.

$$x^{2}-2x+1=16$$

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}.

$$\left(x-1\right)^{2}=16$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$x-1=4 x-1=-4$$

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

$$x=5 x=-3$$