Whakaoti i te 4x^2+48x-81=0

$$4 x ^ { 2 } + 48 x - 81 = 0$$

Whakaoti mō x

$$x = -\frac{27}{2} = -13\frac{1}{2} = -13.5$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

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-81. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=48 ab=4\left(-81\right)=-324$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -324.

$$-1,324 -2,162 -3,108 -4,81 -6,54 -9,36 -12,27 -18,18$$

Tātaihia te tapeke mō ia takirua.

$$-1+324=323 -2+162=160 -3+108=105 -4+81=77 -6+54=48 -9+36=27 -12+27=15 -18+18=0$$

Ko te otinga te takirua ka hoatu i te tapeke 48.

$$a=-6 b=54$$

Tuhia anō te 4x^{2}+48x-81 hei \left(4x^{2}-6x\right)+\left(54x-81\right).

$$\left(4x^{2}-6x\right)+\left(54x-81\right)$$

Tauwehea te 2x i te tuatahi me te 27 i te rōpū tuarua.

$$2x\left(2x-3\right)+27\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(2x+27\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-3=0 me te 2x+27=0.

$$x=\frac{3}{2} x=-\frac{27}{2}$$

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.

$$4x^{2}+48x-81=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 48 mō b, me -81 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 4\left(-81\right)}}{2\times 4}$$

Pūrua 48.

$$x=\frac{-48±\sqrt{2304-4\times 4\left(-81\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-48±\sqrt{2304-16\left(-81\right)}}{2\times 4}$$

Whakareatia -16 ki te -81.

$$x=\frac{-48±\sqrt{2304+1296}}{2\times 4}$$

Tāpiri 2304 ki te 1296.

$$x=\frac{-48±\sqrt{3600}}{2\times 4}$$

Tuhia te pūtakerua o te 3600.

$$x=\frac{-48±60}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-48±60}{8}$$

Nā, me whakaoti te whārite x=\frac{-48±60}{8} ina he tāpiri te ±. Tāpiri -48 ki te 60.

$$x=\frac{12}{8}$$

Whakahekea te hautanga \frac{12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-48±60}{8} ina he tango te ±. Tango 60 mai i -48.

$$x=-\frac{108}{8}$$

Whakahekea te hautanga \frac{-108}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{27}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{3}{2} x=-\frac{27}{2}$$

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.

$$4x^{2}+48x-81=0$$

Me tāpiri 81 ki ngā taha e rua o te whārite.

$$4x^{2}+48x-81-\left(-81\right)=-\left(-81\right)$$

Mā te tango i te -81 i a ia ake anō ka toe ko te 0.

$$4x^{2}+48x=-\left(-81\right)$$

Tango -81 mai i 0.

$$4x^{2}+48x=81$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}+48x}{4}=\frac{81}{4}$$

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

$$x^{2}+\frac{48}{4}x=\frac{81}{4}$$

Whakawehe 48 ki te 4.

$$x^{2}+12x=\frac{81}{4}$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=\frac{81}{4}+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=\frac{81}{4}+36$$

Tāpiri \frac{81}{4} ki te 36.

$$x^{2}+12x+36=\frac{225}{4}$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=\frac{225}{4}$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{\frac{225}{4}}$$

Whakarūnātia.

$$x+6=\frac{15}{2} x+6=-\frac{15}{2}$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=\frac{3}{2} x=-\frac{27}{2}$$