Whakaoti i te 3x+2/9-x^2=5x+1/x+3-1/3-x

$$\frac { 3 x + 2 } { 9 - x ^ { 2 } } = \frac { 5 x + 1 } { x + 3 } - \frac { 1 } { 3 - x }$$

Whakaoti mō x

$$x=\frac{\sqrt{15}}{5}+1\approx 1.774596669$$

$$x=-\frac{\sqrt{15}}{5}+1\approx 0.225403331$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 9-x^{2},x+3,3-x.

$$-\left(3x+2\right)=\left(x-3\right)\left(5x+1\right)+3+x$$

Hei kimi i te tauaro o 3x+2, kimihia te tauaro o ia taurangi.

$$-3x-2=\left(x-3\right)\left(5x+1\right)+3+x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te 5x+1 ka whakakotahi i ngā kupu rite.

$$-3x-2=5x^{2}-14x-3+3+x$$

Tāpirihia te -3 ki te 3, ka 0.

$$-3x-2=5x^{2}-14x+x$$

Pahekotia te -14x me x, ka -13x.

$$-3x-2=5x^{2}-13x$$

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

$$-3x-2-5x^{2}=-13x$$

Me tāpiri te 13x ki ngā taha e rua.

$$-3x-2-5x^{2}+13x=0$$

Pahekotia te -3x me 13x, ka 10x.

$$10x-2-5x^{2}=0$$

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.

$$-5x^{2}+10x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, 10 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\left(-2\right)}}{2\left(-5\right)}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\left(-5\right)\left(-2\right)}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-10±\sqrt{100+20\left(-2\right)}}{2\left(-5\right)}$$

Whakareatia 20 ki te -2.

$$x=\frac{-10±\sqrt{100-40}}{2\left(-5\right)}$$

Tāpiri 100 ki te -40.

$$x=\frac{-10±\sqrt{60}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 60.

$$x=\frac{-10±2\sqrt{15}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{-10±2\sqrt{15}}{-10}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{15}}{-10} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-10}{-10}$$

Whakawehe -10+2\sqrt{15} ki te -10.

$$x=-\frac{\sqrt{15}}{5}+1$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{15}}{-10} ina he tango te ±. Tango 2\sqrt{15} mai i -10.

$$x=\frac{-2\sqrt{15}-10}{-10}$$

Whakawehe -10-2\sqrt{15} ki te -10.

$$x=\frac{\sqrt{15}}{5}+1$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{15}}{5}+1 x=\frac{\sqrt{15}}{5}+1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o 9-x^{2},x+3,3-x.

$$-\left(3x+2\right)=\left(x-3\right)\left(5x+1\right)+3+x$$

Hei kimi i te tauaro o 3x+2, kimihia te tauaro o ia taurangi.

$$-3x-2=\left(x-3\right)\left(5x+1\right)+3+x$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te 5x+1 ka whakakotahi i ngā kupu rite.

$$-3x-2=5x^{2}-14x-3+3+x$$

Tāpirihia te -3 ki te 3, ka 0.

$$-3x-2=5x^{2}-14x+x$$

Pahekotia te -14x me x, ka -13x.

$$-3x-2=5x^{2}-13x$$

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

$$-3x-2-5x^{2}=-13x$$

Me tāpiri te 13x ki ngā taha e rua.

$$-3x-2-5x^{2}+13x=0$$

Pahekotia te -3x me 13x, ka 10x.

$$10x-2-5x^{2}=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$10x-5x^{2}=2$$

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.

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

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5x^{2}+10x}{-5}=\frac{2}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$x^{2}+\frac{10}{-5}x=\frac{2}{-5}$$

Whakawehe 10 ki te -5.

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

Whakawehe 2 ki te -5.

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

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=-\frac{2}{5}+1$$

Tāpiri -\frac{2}{5} ki te 1.

$$x^{2}-2x+1=\frac{3}{5}$$

Tauwehea x^{2}-2x+1. 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}.

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

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{3}{5}}$$

Whakarūnātia.

$$x-1=\frac{\sqrt{15}}{5} x-1=-\frac{\sqrt{15}}{5}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{15}}{5}+1 x=-\frac{\sqrt{15}}{5}+1$$