Whakaoti i te 5x+9/8x-1-5x+1/3x-1=1

$$\frac { 5 x + 9 } { 8 x - 1 } - \frac { 5 x + 1 } { 3 x - 1 } = 1$$

Whakaoti mō x (complex solution)

$$x=\frac{-6\sqrt{6}i+15}{49}\approx 0.306122449-0.29993752i$$

$$x=\frac{15+6\sqrt{6}i}{49}\approx 0.306122449+0.29993752i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara \frac{1}{8},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3x-1\right)\left(8x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 8x-1,3x-1.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 5x+9 ka whakakotahi i ngā kupu rite.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 8x-1 ki te 5x+1 ka whakakotahi i ngā kupu rite.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

Hei kimi i te tauaro o 40x^{2}+3x-1, kimihia te tauaro o ia taurangi.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Pahekotia te 15x^{2} me -40x^{2}, ka -25x^{2}.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Pahekotia te 22x me -3x, ka 19x.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

Tāpirihia te -9 ki te 1, ka -8.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 8x-1 ka whakakotahi i ngā kupu rite.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Tangohia te 24x^{2} mai i ngā taha e rua.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

Pahekotia te -25x^{2} me -24x^{2}, ka -49x^{2}.

$$-49x^{2}+19x-8=-11x+1$$

Me tāpiri te 11x ki ngā taha e rua.

$$-49x^{2}+19x-8+11x=1$$

Pahekotia te 19x me 11x, ka 30x.

$$-49x^{2}+30x-8=1$$

Tangohia te 1 mai i ngā taha e rua.

$$-49x^{2}+30x-8-1=0$$

Tangohia te 1 i te -8, ka -9.

$$-49x^{2}+30x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -49 mō a, 30 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$x=\frac{-30±\sqrt{900+196\left(-9\right)}}{2\left(-49\right)}$$

Whakareatia 196 ki te -9.

$$x=\frac{-30±\sqrt{900-1764}}{2\left(-49\right)}$$

Tāpiri 900 ki te -1764.

$$x=\frac{-30±\sqrt{-864}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te -864.

$$x=\frac{-30±12\sqrt{6}i}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$x=\frac{-30±12\sqrt{6}i}{-98}$$

Nā, me whakaoti te whārite x=\frac{-30±12\sqrt{6}i}{-98} ina he tāpiri te ±. Tāpiri -30 ki te 12i\sqrt{6}.

$$x=\frac{-30+12\sqrt{6}i}{-98}$$

Whakawehe -30+12i\sqrt{6} ki te -98.

$$x=\frac{-6\sqrt{6}i+15}{49}$$

Nā, me whakaoti te whārite x=\frac{-30±12\sqrt{6}i}{-98} ina he tango te ±. Tango 12i\sqrt{6} mai i -30.

$$x=\frac{-12\sqrt{6}i-30}{-98}$$

Whakawehe -30-12i\sqrt{6} ki te -98.

$$x=\frac{15+6\sqrt{6}i}{49}$$

Kua oti te whārite te whakatau.

$$x=\frac{-6\sqrt{6}i+15}{49} x=\frac{15+6\sqrt{6}i}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara \frac{1}{8},\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3x-1\right)\left(8x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 8x-1,3x-1.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 5x+9 ka whakakotahi i ngā kupu rite.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 8x-1 ki te 5x+1 ka whakakotahi i ngā kupu rite.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

Hei kimi i te tauaro o 40x^{2}+3x-1, kimihia te tauaro o ia taurangi.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Pahekotia te 15x^{2} me -40x^{2}, ka -25x^{2}.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

Pahekotia te 22x me -3x, ka 19x.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

Tāpirihia te -9 ki te 1, ka -8.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 8x-1 ka whakakotahi i ngā kupu rite.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Tangohia te 24x^{2} mai i ngā taha e rua.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

Pahekotia te -25x^{2} me -24x^{2}, ka -49x^{2}.

$$-49x^{2}+19x-8=-11x+1$$

Me tāpiri te 11x ki ngā taha e rua.

$$-49x^{2}+19x-8+11x=1$$

Pahekotia te 19x me 11x, ka 30x.

$$-49x^{2}+30x-8=1$$

Me tāpiri te 8 ki ngā taha e rua.

$$-49x^{2}+30x=1+8$$

Tāpirihia te 1 ki te 8, ka 9.

$$-49x^{2}+30x=9$$

Whakawehea ngā taha e rua ki te -49.

$$\frac{-49x^{2}+30x}{-49}=\frac{9}{-49}$$

Mā te whakawehe ki te -49 ka wetekia te whakareanga ki te -49.

$$x^{2}+\frac{30}{-49}x=\frac{9}{-49}$$

Whakawehe 30 ki te -49.

$$x^{2}-\frac{30}{49}x=\frac{9}{-49}$$

Whakawehe 9 ki te -49.

$$x^{2}-\frac{30}{49}x=-\frac{9}{49}$$

Whakawehea te -\frac{30}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{49}. Nā, tāpiria te pūrua o te -\frac{15}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{30}{49}x+\left(-\frac{15}{49}\right)^{2}=-\frac{9}{49}+\left(-\frac{15}{49}\right)^{2}$$

Pūruatia -\frac{15}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{9}{49}+\frac{225}{2401}$$

Tāpiri -\frac{9}{49} ki te \frac{225}{2401} 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.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{216}{2401}$$

Tauwehea x^{2}-\frac{30}{49}x+\frac{225}{2401}. 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}.

$$\left(x-\frac{15}{49}\right)^{2}=-\frac{216}{2401}$$

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

$$\sqrt{\left(x-\frac{15}{49}\right)^{2}}=\sqrt{-\frac{216}{2401}}$$

Whakarūnātia.

$$x-\frac{15}{49}=\frac{6\sqrt{6}i}{49} x-\frac{15}{49}=-\frac{6\sqrt{6}i}{49}$$

Me tāpiri \frac{15}{49} ki ngā taha e rua o te whārite.

$$x=\frac{15+6\sqrt{6}i}{49} x=\frac{-6\sqrt{6}i+15}{49}$$