Whakaoti i te x^2+6x-7/3x^2-x-2=5

$$\frac { x ^ { 2 } + 6 x - 7 } { 3 x ^ { 2 } - x - 2 } = 5$$

Whakaoti mō x

$$x=-\frac{3}{14}\approx -0.214285714$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{2}{3},1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-5 ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Pahekotia te x^{2} me -15x^{2}, ka -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Me tāpiri te 5x ki ngā taha e rua.

$$-14x^{2}+6x-7+5x=-10$$

Pahekotia te 6x me 5x, ka 11x.

$$-14x^{2}+11x-7=-10$$

Me tāpiri te 10 ki ngā taha e rua.

$$-14x^{2}+11x-7+10=0$$

Tāpirihia te -7 ki te 10, ka 3.

$$-14x^{2}+11x+3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -14x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=-14\times 3=-42$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -42.

$$-1,42 -2,21 -3,14 -6,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=14 b=-3$$

Tuhia anō te -14x^{2}+11x+3 hei \left(-14x^{2}+14x\right)+\left(-3x+3\right).

$$\left(-14x^{2}+14x\right)+\left(-3x+3\right)$$

Tauwehea te 14x i te tuatahi me te 3 i te rōpū tuarua.

$$14x\left(-x+1\right)+3\left(-x+1\right)$$

Whakatauwehea atu te kīanga pātahi -x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+1\right)\left(14x+3\right)$$

Hei kimi otinga whārite, me whakaoti te -x+1=0 me te 14x+3=0.

$$x=1 x=-\frac{3}{14}$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=-\frac{3}{14}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{2}{3},1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-5 ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Pahekotia te x^{2} me -15x^{2}, ka -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Me tāpiri te 5x ki ngā taha e rua.

$$-14x^{2}+6x-7+5x=-10$$

Pahekotia te 6x me 5x, ka 11x.

$$-14x^{2}+11x-7=-10$$

Me tāpiri te 10 ki ngā taha e rua.

$$-14x^{2}+11x-7+10=0$$

Tāpirihia te -7 ki te 10, ka 3.

$$-14x^{2}+11x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -14 mō a, 11 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

Whakareatia -4 ki te -14.

$$x=\frac{-11±\sqrt{121+56\times 3}}{2\left(-14\right)}$$

Whakareatia 56 ki te 3.

$$x=\frac{-11±\sqrt{121+168}}{2\left(-14\right)}$$

Tāpiri 121 ki te 168.

$$x=\frac{-11±\sqrt{289}}{2\left(-14\right)}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-11±17}{2\left(-14\right)}$$

Whakareatia 2 ki te -14.

$$x=\frac{-11±17}{-28}$$

Nā, me whakaoti te whārite x=\frac{-11±17}{-28} ina he tāpiri te ±. Tāpiri -11 ki te 17.

$$x=\frac{6}{-28}$$

Whakahekea te hautanga \frac{6}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{3}{14}$$

Nā, me whakaoti te whārite x=\frac{-11±17}{-28} ina he tango te ±. Tango 17 mai i -11.

$$x=-\frac{28}{-28}$$

Whakawehe -28 ki te -28.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=-\frac{3}{14} x=1$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=-\frac{3}{14}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -\frac{2}{3},1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(3x+2\right).

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te x-1.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x-5 ki te 3x+2 ka whakakotahi i ngā kupu rite.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

Pahekotia te x^{2} me -15x^{2}, ka -14x^{2}.

$$-14x^{2}+6x-7=-5x-10$$

Me tāpiri te 5x ki ngā taha e rua.

$$-14x^{2}+6x-7+5x=-10$$

Pahekotia te 6x me 5x, ka 11x.

$$-14x^{2}+11x-7=-10$$

Me tāpiri te 7 ki ngā taha e rua.

$$-14x^{2}+11x=-10+7$$

Tāpirihia te -10 ki te 7, ka -3.

$$-14x^{2}+11x=-3$$

Whakawehea ngā taha e rua ki te -14.

$$\frac{-14x^{2}+11x}{-14}=-\frac{3}{-14}$$

Mā te whakawehe ki te -14 ka wetekia te whakareanga ki te -14.

$$x^{2}+\frac{11}{-14}x=-\frac{3}{-14}$$

Whakawehe 11 ki te -14.

$$x^{2}-\frac{11}{14}x=-\frac{3}{-14}$$

Whakawehe -3 ki te -14.

$$x^{2}-\frac{11}{14}x=\frac{3}{14}$$

Whakawehea te -\frac{11}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{28}. Nā, tāpiria te pūrua o te -\frac{11}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{14}x+\left(-\frac{11}{28}\right)^{2}=\frac{3}{14}+\left(-\frac{11}{28}\right)^{2}$$

Pūruatia -\frac{11}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{3}{14}+\frac{121}{784}$$

Tāpiri \frac{3}{14} ki te \frac{121}{784} 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.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{289}{784}$$

Tauwehea x^{2}-\frac{11}{14}x+\frac{121}{784}. 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}.

$$\left(x-\frac{11}{28}\right)^{2}=\frac{289}{784}$$

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

$$\sqrt{\left(x-\frac{11}{28}\right)^{2}}=\sqrt{\frac{289}{784}}$$

Whakarūnātia.

$$x-\frac{11}{28}=\frac{17}{28} x-\frac{11}{28}=-\frac{17}{28}$$

Me tāpiri \frac{11}{28} ki ngā taha e rua o te whārite.

$$x=1 x=-\frac{3}{14}$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=-\frac{3}{14}$$