Whakaoti i te 3x^2-8x+4/x-2=5x+8

$$\frac { 3 x ^ { 2 } - 8 x + 4 } { x - 2 } = 5 x + 8$$

Whakaoti mō x

$$x=-5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$3x^{2}-8x+4=5x\left(x-2\right)+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x-2.

$$3x^{2}-8x+4=5x^{2}-10x+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 8.

$$3x^{2}-8x+4=5x^{2}-10x+8x-16$$

Pahekotia te -10x me 8x, ka -2x.

$$3x^{2}-8x+4=5x^{2}-2x-16$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$3x^{2}-8x+4-5x^{2}=-2x-16$$

Pahekotia te 3x^{2} me -5x^{2}, ka -2x^{2}.

$$-2x^{2}-8x+4=-2x-16$$

Me tāpiri te 2x ki ngā taha e rua.

$$-2x^{2}-8x+4+2x=-16$$

Pahekotia te -8x me 2x, ka -6x.

$$-2x^{2}-6x+4=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$-2x^{2}-6x+4+16=0$$

Tāpirihia te 4 ki te 16, ka 20.

$$-2x^{2}-6x+20=0$$

Whakawehea ngā taha e rua ki te 2.

$$-x^{2}-3x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-10=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

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

$$a=2 b=-5$$

Tuhia anō te -x^{2}-3x+10 hei \left(-x^{2}+2x\right)+\left(-5x+10\right).

$$\left(-x^{2}+2x\right)+\left(-5x+10\right)$$

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

$$x\left(-x+2\right)+5\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te -x+2=0 me te x+5=0.

$$x=2 x=-5$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$3x^{2}-8x+4=5x\left(x-2\right)+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x-2.

$$3x^{2}-8x+4=5x^{2}-10x+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 8.

$$3x^{2}-8x+4=5x^{2}-10x+8x-16$$

Pahekotia te -10x me 8x, ka -2x.

$$3x^{2}-8x+4=5x^{2}-2x-16$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$3x^{2}-8x+4-5x^{2}=-2x-16$$

Pahekotia te 3x^{2} me -5x^{2}, ka -2x^{2}.

$$-2x^{2}-8x+4=-2x-16$$

Me tāpiri te 2x ki ngā taha e rua.

$$-2x^{2}-8x+4+2x=-16$$

Pahekotia te -8x me 2x, ka -6x.

$$-2x^{2}-6x+4=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$-2x^{2}-6x+4+16=0$$

Tāpirihia te 4 ki te 16, ka 20.

$$-2x^{2}-6x+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, -6 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-\left(-6\right)±\sqrt{36+8\times 20}}{2\left(-2\right)}$$

Whakareatia 8 ki te 20.

$$x=\frac{-\left(-6\right)±\sqrt{36+160}}{2\left(-2\right)}$$

Tāpiri 36 ki te 160.

$$x=\frac{-\left(-6\right)±\sqrt{196}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-\left(-6\right)±14}{2\left(-2\right)}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±14}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{6±14}{-4}$$

Nā, me whakaoti te whārite x=\frac{6±14}{-4} ina he tāpiri te ±. Tāpiri 6 ki te 14.

$$x=\frac{20}{-4}$$

Whakawehe 20 ki te -4.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{6±14}{-4} ina he tango te ±. Tango 14 mai i 6.

$$x=-\frac{8}{-4}$$

Whakawehe -8 ki te -4.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-5 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$3x^{2}-8x+4=5x\left(x-2\right)+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x-2.

$$3x^{2}-8x+4=5x^{2}-10x+\left(x-2\right)\times 8$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 8.

$$3x^{2}-8x+4=5x^{2}-10x+8x-16$$

Pahekotia te -10x me 8x, ka -2x.

$$3x^{2}-8x+4=5x^{2}-2x-16$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$3x^{2}-8x+4-5x^{2}=-2x-16$$

Pahekotia te 3x^{2} me -5x^{2}, ka -2x^{2}.

$$-2x^{2}-8x+4=-2x-16$$

Me tāpiri te 2x ki ngā taha e rua.

$$-2x^{2}-8x+4+2x=-16$$

Pahekotia te -8x me 2x, ka -6x.

$$-2x^{2}-6x+4=-16$$

Tangohia te 4 mai i ngā taha e rua.

$$-2x^{2}-6x=-16-4$$

Tangohia te 4 i te -16, ka -20.

$$-2x^{2}-6x=-20$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}-6x}{-2}=-\frac{20}{-2}$$

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

$$x^{2}+\left(-\frac{6}{-2}\right)x=-\frac{20}{-2}$$

Whakawehe -6 ki te -2.

$$x^{2}+3x=-\frac{20}{-2}$$

Whakawehe -20 ki te -2.

$$x^{2}+3x=10$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

Tauwehea x^{2}+3x+\frac{9}{4}. 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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{7}{2} x+\frac{3}{2}=-\frac{7}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=2 x=-5$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$