Whakaoti i te 2x^3-12x^2+9x/4x^2(x^2+3)=(x-3)/2{x^2+6x}

$$\frac{ 2 { x }^{ 3 } -12 { x }^{ 2 } +9x }{ 4 { x }^{ 2 } ( { x }^{ 2 } +3) } = \frac{ (x-3) }{ 2 { x }^{ 2 } +6x }$$

Whakaoti mō x

$$x = \frac{15}{11} = 1\frac{4}{11} \approx 1.363636364$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,0 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)x^{2}\left(x^{2}+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te 2x^{3}-12x^{2}+9x ka whakakotahi i ngā kupu rite.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x^{3}+6x ki te x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Tangohia te 2x^{4} mai i ngā taha e rua.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Pahekotia te 2x^{4} me -2x^{4}, ka 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Me tāpiri te 6x^{3} ki ngā taha e rua.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Pahekotia te -6x^{3} me 6x^{3}, ka 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-27x^{2}+27x-6x^{2}=-18x$$

Pahekotia te -27x^{2} me -6x^{2}, ka -33x^{2}.

$$-33x^{2}+27x=-18x$$

Me tāpiri te 18x ki ngā taha e rua.

$$-33x^{2}+27x+18x=0$$

Pahekotia te 27x me 18x, ka 45x.

$$-33x^{2}+45x=0$$

Tauwehea te x.

$$x\left(-33x+45\right)=0$$

Hei kimi otinga whārite, me whakaoti te x=0 me te -33x+45=0.

$$x=0 x=\frac{15}{11}$$

Tē taea kia ōrite te tāupe x ki 0.

$$x=\frac{15}{11}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,0 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)x^{2}\left(x^{2}+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te 2x^{3}-12x^{2}+9x ka whakakotahi i ngā kupu rite.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x^{3}+6x ki te x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Tangohia te 2x^{4} mai i ngā taha e rua.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Pahekotia te 2x^{4} me -2x^{4}, ka 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Me tāpiri te 6x^{3} ki ngā taha e rua.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Pahekotia te -6x^{3} me 6x^{3}, ka 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-27x^{2}+27x-6x^{2}=-18x$$

Pahekotia te -27x^{2} me -6x^{2}, ka -33x^{2}.

$$-33x^{2}+27x=-18x$$

Me tāpiri te 18x ki ngā taha e rua.

$$-33x^{2}+27x+18x=0$$

Pahekotia te 27x me 18x, ka 45x.

$$-33x^{2}+45x=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -33 mō a, 45 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}}}{2\left(-33\right)}$$

Tuhia te pūtakerua o te 45^{2}.

$$x=\frac{-45±45}{2\left(-33\right)}$$

Whakareatia 2 ki te -33.

$$x=\frac{-45±45}{-66}$$

Nā, me whakaoti te whārite x=\frac{-45±45}{-66} ina he tāpiri te ±. Tāpiri -45 ki te 45.

$$x=\frac{0}{-66}$$

Whakawehe 0 ki te -66.

$$x=0$$

Nā, me whakaoti te whārite x=\frac{-45±45}{-66} ina he tango te ±. Tango 45 mai i -45.

$$x=-\frac{90}{-66}$$

Whakahekea te hautanga \frac{-90}{-66} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{15}{11}$$

Kua oti te whārite te whakatau.

$$x=0 x=\frac{15}{11}$$

Tē taea kia ōrite te tāupe x ki 0.

$$x=\frac{15}{11}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,0 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)x^{2}\left(x^{2}+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te 2x^{3}-12x^{2}+9x ka whakakotahi i ngā kupu rite.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x^{3}+6x ki te x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Tangohia te 2x^{4} mai i ngā taha e rua.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Pahekotia te 2x^{4} me -2x^{4}, ka 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Me tāpiri te 6x^{3} ki ngā taha e rua.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Pahekotia te -6x^{3} me 6x^{3}, ka 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$-27x^{2}+27x-6x^{2}=-18x$$

Pahekotia te -27x^{2} me -6x^{2}, ka -33x^{2}.

$$-33x^{2}+27x=-18x$$

Me tāpiri te 18x ki ngā taha e rua.

$$-33x^{2}+27x+18x=0$$

Pahekotia te 27x me 18x, ka 45x.

$$-33x^{2}+45x=0$$

Whakawehea ngā taha e rua ki te -33.

$$\frac{-33x^{2}+45x}{-33}=\frac{0}{-33}$$

Mā te whakawehe ki te -33 ka wetekia te whakareanga ki te -33.

$$x^{2}+\frac{45}{-33}x=\frac{0}{-33}$$

Whakahekea te hautanga \frac{45}{-33} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{15}{11}x=\frac{0}{-33}$$

Whakawehe 0 ki te -33.

$$x^{2}-\frac{15}{11}x=0$$

Whakawehea te -\frac{15}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{22}. Nā, tāpiria te pūrua o te -\frac{15}{22} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{15}{11}x+\left(-\frac{15}{22}\right)^{2}=\left(-\frac{15}{22}\right)^{2}$$

Pūruatia -\frac{15}{22} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{15}{11}x+\frac{225}{484}=\frac{225}{484}$$

Tauwehea te x^{2}-\frac{15}{11}x+\frac{225}{484}. 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}.

$$\left(x-\frac{15}{22}\right)^{2}=\frac{225}{484}$$

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

$$\sqrt{\left(x-\frac{15}{22}\right)^{2}}=\sqrt{\frac{225}{484}}$$

Whakarūnātia.

$$x-\frac{15}{22}=\frac{15}{22} x-\frac{15}{22}=-\frac{15}{22}$$

Me tāpiri \frac{15}{22} ki ngā taha e rua o te whārite.

$$x=\frac{15}{11} x=0$$

Tē taea kia ōrite te tāupe x ki 0.

$$x=\frac{15}{11}$$