Whakaoti i te (x+3)(6-x)/36-4x^2=-1/4

$$\frac { ( x + 3 ) ( 6 - x ) } { 36 - 4 x ^ { 2 } } = - \frac { 1 } { 4 }$$

Whakaoti mō x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 36-4x^{2},4.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x+3.

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x-3 ki te 6-x ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x-3.

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x+3 ki te x+3 ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-x^{2}+9$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-3x+x^{2}-18+x^{2}=9$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$-3x+2x^{2}-18=9$$

Tangohia te 9 mai i ngā taha e rua.

$$-3x+2x^{2}-18-9=0$$

Tangohia te 9 i te -18, ka -27.

$$-3x+2x^{2}-27=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$2x^{2}-3x-27=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=2\left(-27\right)=-54$$

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 -54.

$$1,-54 2,-27 3,-18 6,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-54=-53 2-27=-25 3-18=-15 6-9=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-9 b=6$$

Tuhia anō te 2x^{2}-3x-27 hei \left(2x^{2}-9x\right)+\left(6x-27\right).

$$\left(2x^{2}-9x\right)+\left(6x-27\right)$$

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

$$x\left(2x-9\right)+3\left(2x-9\right)$$

Whakatauwehea atu te kīanga pātahi 2x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-9\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-9=0 me te x+3=0.

$$x=\frac{9}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{9}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 36-4x^{2},4.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x+3.

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x-3 ki te 6-x ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x-3.

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x+3 ki te x+3 ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-x^{2}+9$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-3x+x^{2}-18+x^{2}=9$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$-3x+2x^{2}-18=9$$

Tangohia te 9 mai i ngā taha e rua.

$$-3x+2x^{2}-18-9=0$$

Tangohia te 9 i te -18, ka -27.

$$-3x+2x^{2}-27=0$$

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.

$$2x^{2}-3x-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -3 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-27\right)}}{2\times 2}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-27\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-27\right)}}{2\times 2}$$

Whakareatia -8 ki te -27.

$$x=\frac{-\left(-3\right)±\sqrt{9+216}}{2\times 2}$$

Tāpiri 9 ki te 216.

$$x=\frac{-\left(-3\right)±\sqrt{225}}{2\times 2}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-\left(-3\right)±15}{2\times 2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±15}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{3±15}{4}$$

Nā, me whakaoti te whārite x=\frac{3±15}{4} ina he tāpiri te ±. Tāpiri 3 ki te 15.

$$x=\frac{18}{4}$$

Whakahekea te hautanga \frac{18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{9}{2}$$

Nā, me whakaoti te whārite x=\frac{3±15}{4} ina he tango te ±. Tango 15 mai i 3.

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

Whakawehe -12 ki te 4.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=\frac{9}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 36-4x^{2},4.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x+3.

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x-3 ki te 6-x ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te x-3.

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -x+3 ki te x+3 ka whakakotahi i ngā kupu rite.

$$-3x+x^{2}-18=-x^{2}+9$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$-3x+x^{2}-18+x^{2}=9$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$-3x+2x^{2}-18=9$$

Me tāpiri te 18 ki ngā taha e rua.

$$-3x+2x^{2}=9+18$$

Tāpirihia te 9 ki te 18, ka 27.

$$-3x+2x^{2}=27$$

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.

$$2x^{2}-3x=27$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{27}{2}+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{27}{2}+\frac{9}{16}$$

Tāpiri \frac{27}{2} ki te \frac{9}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{225}{16}$$

Tauwehea te x^{2}-\frac{3}{2}x+\frac{9}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{225}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{15}{4} x-\frac{3}{4}=-\frac{15}{4}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

$$x=\frac{9}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{9}{2}$$