Whakaoti i te x^2-5x+6/x^2-7x+12=2

$$\frac{ { x }^{ 2 } -5x+6 }{ { x }^{ 2 } -7x+12 } =2$$

Whakaoti mō x

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right).

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-8 ki te x-3 ka whakakotahi i ngā kupu rite.

$$x^{2}-5x+6=2x^{2}-14x+24$$

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

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

Me tāpiri te 14x ki ngā taha e rua.

$$-x^{2}-5x+6+14x=24$$

Pahekotia te -5x me 14x, ka 9x.

$$-x^{2}+9x+6=24$$

Tangohia te 24 mai i ngā taha e rua.

$$-x^{2}+9x+6-24=0$$

Tangohia te 24 i te 6, ka -18.

$$-x^{2}+9x-18=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=-\left(-18\right)=18$$

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 18.

$$1,18 2,9 3,6$$

Tātaihia te tapeke mō ia takirua.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=6 b=3$$

Tuhia anō te -x^{2}+9x-18 hei \left(-x^{2}+6x\right)+\left(3x-18\right).

$$\left(-x^{2}+6x\right)+\left(3x-18\right)$$

Tauwehea te -x i te tuatahi me te 3 i te rōpū tuarua.

$$-x\left(x-6\right)+3\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(-x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te -x+3=0.

$$x=6 x=3$$

Tē taea kia ōrite te tāupe x ki 3.

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right).

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-8 ki te x-3 ka whakakotahi i ngā kupu rite.

$$x^{2}-5x+6=2x^{2}-14x+24$$

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

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

Me tāpiri te 14x ki ngā taha e rua.

$$-x^{2}-5x+6+14x=24$$

Pahekotia te -5x me 14x, ka 9x.

$$-x^{2}+9x+6=24$$

Tangohia te 24 mai i ngā taha e rua.

$$-x^{2}+9x+6-24=0$$

Tangohia te 24 i te 6, ka -18.

$$-x^{2}+9x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-9±\sqrt{81+4\left(-18\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -18.

$$x=\frac{-9±\sqrt{81-72}}{2\left(-1\right)}$$

Tāpiri 81 ki te -72.

$$x=\frac{-9±\sqrt{9}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-9±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-9±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{-9±3}{-2} ina he tāpiri te ±. Tāpiri -9 ki te 3.

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

Whakawehe -6 ki te -2.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{-9±3}{-2} ina he tango te ±. Tango 3 mai i -9.

$$x=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=3 x=6$$

Tē taea kia ōrite te tāupe x ki 3.

$$x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,4 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-3\right).

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-4.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-8 ki te x-3 ka whakakotahi i ngā kupu rite.

$$x^{2}-5x+6=2x^{2}-14x+24$$

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

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

Me tāpiri te 14x ki ngā taha e rua.

$$-x^{2}-5x+6+14x=24$$

Pahekotia te -5x me 14x, ka 9x.

$$-x^{2}+9x+6=24$$

Tangohia te 6 mai i ngā taha e rua.

$$-x^{2}+9x=24-6$$

Tangohia te 6 i te 24, ka 18.

$$-x^{2}+9x=18$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+9x}{-1}=\frac{18}{-1}$$

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

$$x^{2}+\frac{9}{-1}x=\frac{18}{-1}$$

Whakawehe 9 ki te -1.

$$x^{2}-9x=\frac{18}{-1}$$

Whakawehe 18 ki te -1.

$$x^{2}-9x=-18$$

Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-18+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=-18+\frac{81}{4}$$

Tāpiri -18 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{9}{4}$$

Tauwehea x^{2}-9x+\frac{81}{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}.

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

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

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

Whakarūnātia.

$$x-\frac{9}{2}=\frac{3}{2} x-\frac{9}{2}=-\frac{3}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=6 x=3$$

Tē taea kia ōrite te tāupe x ki 3.

$$x=6$$