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

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

Whakaoti mō x

$$x=4$$

$$x = \frac{11}{2} = 5\frac{1}{2} = 5.5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,2 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+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-2,x+5.

$$\left(x+5\right)\left(3x-8\right)=\left(x-2\right)\left(5x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 3x-8 ka whakakotahi i ngā kupu rite.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

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

$$3x^{2}+7x-40=5x^{2}-12x+4$$

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

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

Me tāpiri te 12x ki ngā taha e rua.

$$-2x^{2}+7x-40+12x=4$$

Pahekotia te 7x me 12x, ka 19x.

$$-2x^{2}+19x-40=4$$

Tangohia te 4 mai i ngā taha e rua.

$$-2x^{2}+19x-40-4=0$$

Tangohia te 4 i te -40, ka -44.

$$-2x^{2}+19x-44=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 19 mō b, me -44 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Pūrua 19.

$$x=\frac{-19±\sqrt{361-4\left(-2\right)\left(-44\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-19±\sqrt{361+8\left(-44\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -44.

$$x=\frac{-19±\sqrt{361-352}}{2\left(-2\right)}$$

Tāpiri 361 ki te -352.

$$x=\frac{-19±\sqrt{9}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-19±3}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-19±3}{-4}$$

Nā, me whakaoti te whārite x=\frac{-19±3}{-4} ina he tāpiri te ±. Tāpiri -19 ki te 3.

$$x=-\frac{16}{-4}$$

Whakawehe -16 ki te -4.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{-19±3}{-4} ina he tango te ±. Tango 3 mai i -19.

$$x=-\frac{22}{-4}$$

Whakahekea te hautanga \frac{-22}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{11}{2}$$

Kua oti te whārite te whakatau.

$$x=4 x=\frac{11}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,2 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+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-2,x+5.

$$\left(x+5\right)\left(3x-8\right)=\left(x-2\right)\left(5x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+5 ki te 3x-8 ka whakakotahi i ngā kupu rite.

$$3x^{2}+7x-40=\left(x-2\right)\left(5x-2\right)$$

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

$$3x^{2}+7x-40=5x^{2}-12x+4$$

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

$$3x^{2}+7x-40-5x^{2}=-12x+4$$

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

$$-2x^{2}+7x-40=-12x+4$$

Me tāpiri te 12x ki ngā taha e rua.

$$-2x^{2}+7x-40+12x=4$$

Pahekotia te 7x me 12x, ka 19x.

$$-2x^{2}+19x-40=4$$

Me tāpiri te 40 ki ngā taha e rua.

$$-2x^{2}+19x=4+40$$

Tāpirihia te 4 ki te 40, ka 44.

$$-2x^{2}+19x=44$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+19x}{-2}=\frac{44}{-2}$$

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

$$x^{2}+\frac{19}{-2}x=\frac{44}{-2}$$

Whakawehe 19 ki te -2.

$$x^{2}-\frac{19}{2}x=\frac{44}{-2}$$

Whakawehe 44 ki te -2.

$$x^{2}-\frac{19}{2}x=-22$$

Whakawehea te -\frac{19}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{4}. Nā, tāpiria te pūrua o te -\frac{19}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{19}{2}x+\left(-\frac{19}{4}\right)^{2}=-22+\left(-\frac{19}{4}\right)^{2}$$

Pūruatia -\frac{19}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=-22+\frac{361}{16}$$

Tāpiri -22 ki te \frac{361}{16}.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=\frac{9}{16}$$

Tauwehea x^{2}-\frac{19}{2}x+\frac{361}{16}. 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}.

$$\left(x-\frac{19}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(x-\frac{19}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

$$x-\frac{19}{4}=\frac{3}{4} x-\frac{19}{4}=-\frac{3}{4}$$

Me tāpiri \frac{19}{4} ki ngā taha e rua o te whārite.

$$x=\frac{11}{2} x=4$$