Whakaoti i te 4x^2-8x-5=0

$$4 x ^ { 2 } - 8 x - 5 = 0$$

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

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 4x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=4\left(-5\right)=-20$$

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 -20.

$$1,-20 2,-10 4,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-20=-19 2-10=-8 4-5=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-10 b=2$$

Tuhia anō te 4x^{2}-8x-5 hei \left(4x^{2}-10x\right)+\left(2x-5\right).

$$\left(4x^{2}-10x\right)+\left(2x-5\right)$$

Whakatauwehea atu 2x i te 4x^{2}-10x.

$$2x\left(2x-5\right)+2x-5$$

Whakatauwehea atu te kīanga pātahi 2x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-5\right)\left(2x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-5=0 me te 2x+1=0.

$$x=\frac{5}{2} x=-\frac{1}{2}$$

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.

$$4x^{2}-8x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -8 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 4\left(-5\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-8\right)±\sqrt{64-16\left(-5\right)}}{2\times 4}$$

Whakareatia -16 ki te -5.

$$x=\frac{-\left(-8\right)±\sqrt{64+80}}{2\times 4}$$

Tāpiri 64 ki te 80.

$$x=\frac{-\left(-8\right)±\sqrt{144}}{2\times 4}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-8\right)±12}{2\times 4}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±12}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{8±12}{8}$$

Nā, me whakaoti te whārite x=\frac{8±12}{8} ina he tāpiri te ±. Tāpiri 8 ki te 12.

$$x=\frac{20}{8}$$

Whakahekea te hautanga \frac{20}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{5}{2}$$

Nā, me whakaoti te whārite x=\frac{8±12}{8} ina he tango te ±. Tango 12 mai i 8.

$$x=-\frac{4}{8}$$

Whakahekea te hautanga \frac{-4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{2} x=-\frac{1}{2}$$

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.

$$4x^{2}-8x-5=0$$

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

$$4x^{2}-8x-5-\left(-5\right)=-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$4x^{2}-8x=-\left(-5\right)$$

Tango -5 mai i 0.

$$4x^{2}-8x=5$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-8x}{4}=\frac{5}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\left(-\frac{8}{4}\right)x=\frac{5}{4}$$

Whakawehe -8 ki te 4.

$$x^{2}-2x=\frac{5}{4}$$

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.

$$x^{2}-2x+1=\frac{5}{4}+1$$

Tāpiri \frac{5}{4} ki te 1.

$$x^{2}-2x+1=\frac{9}{4}$$

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}.

$$\left(x-1\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

$$x-1=\frac{3}{2} x-1=-\frac{3}{2}$$

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

$$x=\frac{5}{2} x=-\frac{1}{2}$$