Whakaoti i te 10/(x+3)(x-7)-8/(x+3)(x-5)=x+10/(x-5)(x-7)

$$\frac { 10 } { ( x + 3 ) ( x - 7 ) } - \frac { 8 } { ( x + 3 ) ( x - 5 ) } = \frac { x + 10 } { ( x - 5 ) ( x - 7 ) }$$

Whakaoti mō x

$$x=-8$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,5,7 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-7\right)\left(x-5\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right).

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 10.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-7 ki te 8.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

Hei kimi i te tauaro o 8x-56, kimihia te tauaro o ia taurangi.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

Pahekotia te 10x me -8x, ka 2x.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

Tāpirihia te -50 ki te 56, ka 6.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te x+10 ka whakakotahi i ngā kupu rite.

$$2x+6=x^{2}+13x+30$$

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

$$2x+6-x^{2}=13x+30$$

Tangohia te 13x mai i ngā taha e rua.

$$2x+6-x^{2}-13x=30$$

Pahekotia te 2x me -13x, ka -11x.

$$-11x+6-x^{2}=30$$

Tangohia te 30 mai i ngā taha e rua.

$$-11x+6-x^{2}-30=0$$

Tangohia te 30 i te 6, ka -24.

$$-11x-24-x^{2}=0$$

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.

$$-x^{2}-11x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -11 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-11\right)±\sqrt{121+4\left(-24\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -24.

$$x=\frac{-\left(-11\right)±\sqrt{121-96}}{2\left(-1\right)}$$

Tāpiri 121 ki te -96.

$$x=\frac{-\left(-11\right)±\sqrt{25}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-11\right)±5}{2\left(-1\right)}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±5}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{11±5}{-2}$$

Nā, me whakaoti te whārite x=\frac{11±5}{-2} ina he tāpiri te ±. Tāpiri 11 ki te 5.

$$x=\frac{16}{-2}$$

Whakawehe 16 ki te -2.

$$x=-8$$

Nā, me whakaoti te whārite x=\frac{11±5}{-2} ina he tango te ±. Tango 5 mai i 11.

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

Whakawehe 6 ki te -2.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=-8 x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=-8$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,5,7 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-7\right)\left(x-5\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o \left(x+3\right)\left(x-7\right),\left(x+3\right)\left(x-5\right),\left(x-5\right)\left(x-7\right).

$$\left(x-5\right)\times 10-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 10.

$$10x-50-\left(x-7\right)\times 8=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-7 ki te 8.

$$10x-50-\left(8x-56\right)=\left(x+3\right)\left(x+10\right)$$

Hei kimi i te tauaro o 8x-56, kimihia te tauaro o ia taurangi.

$$10x-50-8x+56=\left(x+3\right)\left(x+10\right)$$

Pahekotia te 10x me -8x, ka 2x.

$$2x-50+56=\left(x+3\right)\left(x+10\right)$$

Tāpirihia te -50 ki te 56, ka 6.

$$2x+6=\left(x+3\right)\left(x+10\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+3 ki te x+10 ka whakakotahi i ngā kupu rite.

$$2x+6=x^{2}+13x+30$$

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

$$2x+6-x^{2}=13x+30$$

Tangohia te 13x mai i ngā taha e rua.

$$2x+6-x^{2}-13x=30$$

Pahekotia te 2x me -13x, ka -11x.

$$-11x+6-x^{2}=30$$

Tangohia te 6 mai i ngā taha e rua.

$$-11x-x^{2}=30-6$$

Tangohia te 6 i te 30, ka 24.

$$-11x-x^{2}=24$$

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.

$$-x^{2}-11x=24$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-11x}{-1}=\frac{24}{-1}$$

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

$$x^{2}+\left(-\frac{11}{-1}\right)x=\frac{24}{-1}$$

Whakawehe -11 ki te -1.

$$x^{2}+11x=\frac{24}{-1}$$

Whakawehe 24 ki te -1.

$$x^{2}+11x=-24$$

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

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-24+\left(\frac{11}{2}\right)^{2}$$

Pūruatia \frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+11x+\frac{121}{4}=-24+\frac{121}{4}$$

Tāpiri -24 ki te \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=\frac{25}{4}$$

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

$$\left(x+\frac{11}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x+\frac{11}{2}=\frac{5}{2} x+\frac{11}{2}=-\frac{5}{2}$$

Me tango \frac{11}{2} mai i ngā taha e rua o te whārite.

$$x=-3 x=-8$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=-8$$