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

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

Whakaoti mō x

$$x = \frac{\sqrt{155} + 3}{4} \approx 3.862474899$$

$$x=\frac{3-\sqrt{155}}{4}\approx -2.362474899$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{1}{2},\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(2x-1\right)\left(2x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 1-4x^{2},4.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te x+3.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -4x-12 ki te 6-x ka whakakotahi i ngā kupu rite.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te 2x-1.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+1 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$-12x+4x^{2}-72=-4x^{2}+1$$

Me tāpiri te 4x^{2} ki ngā taha e rua.

$$-12x+4x^{2}-72+4x^{2}=1$$

Pahekotia te 4x^{2} me 4x^{2}, ka 8x^{2}.

$$-12x+8x^{2}-72=1$$

Tangohia te 1 mai i ngā taha e rua.

$$-12x+8x^{2}-72-1=0$$

Tangohia te 1 i te -72, ka -73.

$$-12x+8x^{2}-73=0$$

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.

$$8x^{2}-12x-73=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -12 mō b, me -73 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 8\left(-73\right)}}{2\times 8}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 8\left(-73\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-12\right)±\sqrt{144-32\left(-73\right)}}{2\times 8}$$

Whakareatia -32 ki te -73.

$$x=\frac{-\left(-12\right)±\sqrt{144+2336}}{2\times 8}$$

Tāpiri 144 ki te 2336.

$$x=\frac{-\left(-12\right)±\sqrt{2480}}{2\times 8}$$

Tuhia te pūtakerua o te 2480.

$$x=\frac{-\left(-12\right)±4\sqrt{155}}{2\times 8}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4\sqrt{155}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{12±4\sqrt{155}}{16}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{155}}{16} ina he tāpiri te ±. Tāpiri 12 ki te 4\sqrt{155}.

$$x=\frac{4\sqrt{155}+12}{16}$$

Whakawehe 12+4\sqrt{155} ki te 16.

$$x=\frac{\sqrt{155}+3}{4}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{155}}{16} ina he tango te ±. Tango 4\sqrt{155} mai i 12.

$$x=\frac{12-4\sqrt{155}}{16}$$

Whakawehe 12-4\sqrt{155} ki te 16.

$$x=\frac{3-\sqrt{155}}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{1}{2},\frac{1}{2} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(2x-1\right)\left(2x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 1-4x^{2},4.

$$-4\left(x+3\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te x+3.

$$\left(-4x-12\right)\left(6-x\right)=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -4x-12 ki te 6-x ka whakakotahi i ngā kupu rite.

$$-12x+4x^{2}-72=-\left(2x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te 2x-1.

$$-12x+4x^{2}-72=\left(-2x+1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -2x+1 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$-12x+4x^{2}-72=-4x^{2}+1$$

Me tāpiri te 4x^{2} ki ngā taha e rua.

$$-12x+4x^{2}-72+4x^{2}=1$$

Pahekotia te 4x^{2} me 4x^{2}, ka 8x^{2}.

$$-12x+8x^{2}-72=1$$

Me tāpiri te 72 ki ngā taha e rua.

$$-12x+8x^{2}=1+72$$

Tāpirihia te 1 ki te 72, ka 73.

$$-12x+8x^{2}=73$$

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.

$$8x^{2}-12x=73$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-12x}{8}=\frac{73}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\left(-\frac{12}{8}\right)x=\frac{73}{8}$$

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

$$x^{2}-\frac{3}{2}x=\frac{73}{8}$$

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.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{73}{8}+\left(-\frac{3}{4}\right)^{2}$$

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

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{73}{8}+\frac{9}{16}$$

Tāpiri \frac{73}{8} 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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{155}{16}$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{16}. 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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{155}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{155}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{\sqrt{155}}{4} x-\frac{3}{4}=-\frac{\sqrt{155}}{4}$$

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

$$x=\frac{\sqrt{155}+3}{4} x=\frac{3-\sqrt{155}}{4}$$