Whakaoti i te x-1/x+2=10/3x-2

$$\frac { x - 1 } { x + 2 } = \frac { 10 } { 3 x - 2 }$$

Whakaoti mō x

$$x=-1$$

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,\frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+2,3x-2.

$$\left(3x-2\right)\left(x-1\right)=\left(x+2\right)\times 10$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-2 ki te x-1 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 10.

$$3x^{2}-5x+2=10x+20$$

Tangohia te 10x mai i ngā taha e rua.

$$3x^{2}-5x+2-10x=20$$

Pahekotia te -5x me -10x, ka -15x.

$$3x^{2}-15x+2=20$$

Tangohia te 20 mai i ngā taha e rua.

$$3x^{2}-15x+2-20=0$$

Tangohia te 20 i te 2, ka -18.

$$3x^{2}-15x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -15 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\left(-18\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\left(-18\right)}}{2\times 3}$$

Whakareatia -12 ki te -18.

$$x=\frac{-\left(-15\right)±\sqrt{225+216}}{2\times 3}$$

Tāpiri 225 ki te 216.

$$x=\frac{-\left(-15\right)±\sqrt{441}}{2\times 3}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-15\right)±21}{2\times 3}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±21}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{15±21}{6}$$

Nā, me whakaoti te whārite x=\frac{15±21}{6} ina he tāpiri te ±. Tāpiri 15 ki te 21.

$$x=\frac{36}{6}$$

Whakawehe 36 ki te 6.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{15±21}{6} ina he tango te ±. Tango 21 mai i 15.

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

Whakawehe -6 ki te 6.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=6 x=-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,\frac{2}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(3x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+2,3x-2.

$$\left(3x-2\right)\left(x-1\right)=\left(x+2\right)\times 10$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-2 ki te x-1 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 10.

$$3x^{2}-5x+2=10x+20$$

Tangohia te 10x mai i ngā taha e rua.

$$3x^{2}-5x+2-10x=20$$

Pahekotia te -5x me -10x, ka -15x.

$$3x^{2}-15x+2=20$$

Tangohia te 2 mai i ngā taha e rua.

$$3x^{2}-15x=20-2$$

Tangohia te 2 i te 20, ka 18.

$$3x^{2}-15x=18$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-15x}{3}=\frac{18}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\left(-\frac{15}{3}\right)x=\frac{18}{3}$$

Whakawehe -15 ki te 3.

$$x^{2}-5x=\frac{18}{3}$$

Whakawehe 18 ki te 3.

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

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}.

$$x^{2}-5x+\frac{25}{4}=\frac{49}{4}$$

Tauwehea te x^{2}-5x+\frac{25}{4}. 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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{7}{2} x-\frac{5}{2}=-\frac{7}{2}$$

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

$$x=6 x=-1$$