Whakaoti i te 1/1frac{7{9}x}+1/x+1/1frac{7{9}x+2}=1/12

$$\frac { 1 } { 1 \frac { 7 } { 9 } x } + \frac { 1 } { x } + \frac { 1 } { 1 \frac { 7 } { 9 } x + 2 } = \frac { 1 } { 12 }$$

Whakaoti mō x

$$x = \frac{15 \sqrt{193} + 195}{16} \approx 25.21166624$$

$$x=\frac{195-15\sqrt{193}}{16}\approx -0.83666624$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x, arā, te tauraro pātahi he tino iti rawa te kitea o x,12.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

Tāpirihia te \frac{27}{4} ki te 12, ka \frac{75}{4}.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Tangohia te x mai i ngā taha e rua.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Whakaraupapatia anō ngā kīanga tau.

$$-x+54\times \frac{1}{8x+9}x+\frac{75}{4}=0$$

Tē taea kia ōrite te tāupe x ki -\frac{9}{8} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(8x+9\right), arā, te tauraro pātahi he tino iti rawa te kitea o 8x+9,4.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Whakareatia te -1 ki te 4, ka -4.

$$-4x\left(8x+9\right)+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4x ki te 8x+9.

$$-32x^{2}-36x+54\times 4\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Whakareatia te 54 ki te 4, ka 216.

$$-32x^{2}-36x+216\times 1x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Whakareatia te 216 ki te 1, ka 216.

$$-32x^{2}-36x+216x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Pahekotia te -36x me 216x, ka 180x.

$$-32x^{2}+180x+4\left(8x+9\right)\times \frac{75}{4}=0$$

Whakareatia te 4 ki te \frac{75}{4}, ka 75.

$$-32x^{2}+180x+75\left(8x+9\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 75 ki te 8x+9.

$$-32x^{2}+180x+600x+675=0$$

Pahekotia te 180x me 600x, ka 780x.

$$-32x^{2}+780x+675=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -32 mō a, 780 mō b, me 675 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-780±\sqrt{780^{2}-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

Pūrua 780.

$$x=\frac{-780±\sqrt{608400-4\left(-32\right)\times 675}}{2\left(-32\right)}$$

Whakareatia -4 ki te -32.

$$x=\frac{-780±\sqrt{608400+128\times 675}}{2\left(-32\right)}$$

Whakareatia 128 ki te 675.

$$x=\frac{-780±\sqrt{608400+86400}}{2\left(-32\right)}$$

Tāpiri 608400 ki te 86400.

$$x=\frac{-780±\sqrt{694800}}{2\left(-32\right)}$$

Tuhia te pūtakerua o te 694800.

$$x=\frac{-780±60\sqrt{193}}{2\left(-32\right)}$$

Whakareatia 2 ki te -32.

$$x=\frac{-780±60\sqrt{193}}{-64}$$

Nā, me whakaoti te whārite x=\frac{-780±60\sqrt{193}}{-64} ina he tāpiri te ±. Tāpiri -780 ki te 60\sqrt{193}.

$$x=\frac{60\sqrt{193}-780}{-64}$$

Whakawehe -780+60\sqrt{193} ki te -64.

$$x=\frac{195-15\sqrt{193}}{16}$$

Nā, me whakaoti te whārite x=\frac{-780±60\sqrt{193}}{-64} ina he tango te ±. Tango 60\sqrt{193} mai i -780.

$$x=\frac{-60\sqrt{193}-780}{-64}$$

Whakawehe -780-60\sqrt{193} ki te -64.

$$x=\frac{15\sqrt{193}+195}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{195-15\sqrt{193}}{16} x=\frac{15\sqrt{193}+195}{16}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x, arā, te tauraro pātahi he tino iti rawa te kitea o x,12.

$$\frac{27}{4}+12+54x\left(8x+9\right)^{-1}=x$$

Tāpirihia te \frac{27}{4} ki te 12, ka \frac{75}{4}.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}=x$$

Tangohia te x mai i ngā taha e rua.

$$\frac{75}{4}+54x\left(8x+9\right)^{-1}-x=0$$

Tangohia te \frac{75}{4} mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$54x\left(8x+9\right)^{-1}-x=-\frac{75}{4}$$

Whakaraupapatia anō ngā kīanga tau.

$$-x+54\times \frac{1}{8x+9}x=-\frac{75}{4}$$

Tē taea kia ōrite te tāupe x ki -\frac{9}{8} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(8x+9\right), arā, te tauraro pātahi he tino iti rawa te kitea o 8x+9,4.

$$-x\times 4\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

Whakareatia te -1 ki te 4, ka -4.

$$-4x\left(8x+9\right)+54\times 4\times 1x=-75\left(8x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4x ki te 8x+9.

$$-32x^{2}-36x+54\times 4\times 1x=-75\left(8x+9\right)$$

Whakareatia te 54 ki te 4, ka 216.

$$-32x^{2}-36x+216\times 1x=-75\left(8x+9\right)$$

Whakareatia te 216 ki te 1, ka 216.

$$-32x^{2}-36x+216x=-75\left(8x+9\right)$$

Pahekotia te -36x me 216x, ka 180x.

$$-32x^{2}+180x=-75\left(8x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -75 ki te 8x+9.

$$-32x^{2}+180x=-600x-675$$

Me tāpiri te 600x ki ngā taha e rua.

$$-32x^{2}+180x+600x=-675$$

Pahekotia te 180x me 600x, ka 780x.

$$-32x^{2}+780x=-675$$

Whakawehea ngā taha e rua ki te -32.

$$\frac{-32x^{2}+780x}{-32}=-\frac{675}{-32}$$

Mā te whakawehe ki te -32 ka wetekia te whakareanga ki te -32.

$$x^{2}+\frac{780}{-32}x=-\frac{675}{-32}$$

Whakahekea te hautanga \frac{780}{-32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{195}{8}x=-\frac{675}{-32}$$

Whakawehe -675 ki te -32.

$$x^{2}-\frac{195}{8}x=\frac{675}{32}$$

Whakawehea te -\frac{195}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{195}{16}. Nā, tāpiria te pūrua o te -\frac{195}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{195}{8}x+\left(-\frac{195}{16}\right)^{2}=\frac{675}{32}+\left(-\frac{195}{16}\right)^{2}$$

Pūruatia -\frac{195}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{675}{32}+\frac{38025}{256}$$

Tāpiri \frac{675}{32} ki te \frac{38025}{256} 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.

$$x^{2}-\frac{195}{8}x+\frac{38025}{256}=\frac{43425}{256}$$

Tauwehea x^{2}-\frac{195}{8}x+\frac{38025}{256}. 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}.

$$\left(x-\frac{195}{16}\right)^{2}=\frac{43425}{256}$$

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

$$\sqrt{\left(x-\frac{195}{16}\right)^{2}}=\sqrt{\frac{43425}{256}}$$

Whakarūnātia.

$$x-\frac{195}{16}=\frac{15\sqrt{193}}{16} x-\frac{195}{16}=-\frac{15\sqrt{193}}{16}$$

Me tāpiri \frac{195}{16} ki ngā taha e rua o te whārite.

$$x=\frac{15\sqrt{193}+195}{16} x=\frac{195-15\sqrt{193}}{16}$$