Whakaoti i te 3/x^2+2x-8-4/x^2+4x-12=x-4/x^2+10x+24

$$\frac { 3 } { x ^ { 2 } + 2 x - 8 } - \frac { 4 } { x ^ { 2 } + 4 x - 12 } = \frac { x - 4 } { x ^ { 2 } + 10 x + 24 }$$

Whakaoti mō x

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,-4,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+4\right)\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 3.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

Hei kimi i te tauaro o 4x+16, kimihia te tauaro o ia taurangi.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

Pahekotia te 3x me -4x, ka -x.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

Tangohia te 16 i te 18, ka 2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te x-2 ki te x-4 ka whakakotahi i ngā kupu rite.

$$-x+2=x^{2}-6x+8$$

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

$$-x+2-x^{2}=-6x+8$$

Me tāpiri te 6x ki ngā taha e rua.

$$-x+2-x^{2}+6x=8$$

Pahekotia te -x me 6x, ka 5x.

$$5x+2-x^{2}=8$$

Tangohia te 8 mai i ngā taha e rua.

$$5x+2-x^{2}-8=0$$

Tangohia te 8 i te 2, ka -6.

$$5x-6-x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+5x-6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=-\left(-6\right)=6$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 6.

$$1,6 2,3$$

Tātaihia te tapeke mō ia takirua.

$$1+6=7 2+3=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=3 b=2$$

Tuhia anō te -x^{2}+5x-6 hei \left(-x^{2}+3x\right)+\left(2x-6\right).

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

Tauwehea te -x i te tuatahi me te 2 i te rōpū tuarua.

$$-x\left(x-3\right)+2\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(-x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te -x+2=0.

$$x=3 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,-4,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+4\right)\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 3.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

Hei kimi i te tauaro o 4x+16, kimihia te tauaro o ia taurangi.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

Pahekotia te 3x me -4x, ka -x.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

Tangohia te 16 i te 18, ka 2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te x-2 ki te x-4 ka whakakotahi i ngā kupu rite.

$$-x+2=x^{2}-6x+8$$

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

$$-x+2-x^{2}=-6x+8$$

Me tāpiri te 6x ki ngā taha e rua.

$$-x+2-x^{2}+6x=8$$

Pahekotia te -x me 6x, ka 5x.

$$5x+2-x^{2}=8$$

Tangohia te 8 mai i ngā taha e rua.

$$5x+2-x^{2}-8=0$$

Tangohia te 8 i te 2, ka -6.

$$5x-6-x^{2}=0$$

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.

$$-x^{2}+5x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 5 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-5±\sqrt{25+4\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -6.

$$x=\frac{-5±\sqrt{25-24}}{2\left(-1\right)}$$

Tāpiri 25 ki te -24.

$$x=\frac{-5±\sqrt{1}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-5±1}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-5±1}{-2}$$

Nā, me whakaoti te whārite x=\frac{-5±1}{-2} ina he tāpiri te ±. Tāpiri -5 ki te 1.

$$x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-5±1}{-2} ina he tango te ±. Tango 1 mai i -5.

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

Whakawehe -6 ki te -2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=2 x=3$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,-4,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+4\right)\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+2x-8,x^{2}+4x-12,x^{2}+10x+24.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 3.

$$3x+18-\left(x+4\right)\times 4=\left(x-2\right)\left(x-4\right)$$

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

$$3x+18-\left(4x+16\right)=\left(x-2\right)\left(x-4\right)$$

Hei kimi i te tauaro o 4x+16, kimihia te tauaro o ia taurangi.

$$3x+18-4x-16=\left(x-2\right)\left(x-4\right)$$

Pahekotia te 3x me -4x, ka -x.

$$-x+18-16=\left(x-2\right)\left(x-4\right)$$

Tangohia te 16 i te 18, ka 2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te x-2 ki te x-4 ka whakakotahi i ngā kupu rite.

$$-x+2=x^{2}-6x+8$$

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

$$-x+2-x^{2}=-6x+8$$

Me tāpiri te 6x ki ngā taha e rua.

$$-x+2-x^{2}+6x=8$$

Pahekotia te -x me 6x, ka 5x.

$$5x+2-x^{2}=8$$

Tangohia te 2 mai i ngā taha e rua.

$$5x-x^{2}=8-2$$

Tangohia te 2 i te 8, ka 6.

$$5x-x^{2}=6$$

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.

$$-x^{2}+5x=6$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+5x}{-1}=\frac{6}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{5}{-1}x=\frac{6}{-1}$$

Whakawehe 5 ki te -1.

$$x^{2}-5x=\frac{6}{-1}$$

Whakawehe 6 ki te -1.

$$x^{2}-5x=-6$$

Whakawehea te -5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{2}. Nā, tāpiria te pūrua o te -\frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-6+\left(-\frac{5}{2}\right)^{2}$$

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=-6+\frac{25}{4}$$

Tāpiri -6 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{4}$$

Tauwehea x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{1}{2} x-\frac{5}{2}=-\frac{1}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=3 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=3$$