Whakaoti i te 5/2+5/x-2=6/x^2-4

$$\frac { 5 } { 2 } + \frac { 5 } { x - 2 } = \frac { 6 } { x ^ { 2 } - 4 }$$

Whakaoti mō x

$$x=\frac{\sqrt{85}}{5}-1\approx 0.843908891$$

$$x=-\frac{\sqrt{85}}{5}-1\approx -2.843908891$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2,x-2,x^{2}-4.

$$2\left(x-2\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-2.

$$\left(2x-4\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-4 ki te x+2 ka whakakotahi i ngā kupu rite.

$$\left(2x^{2}-8\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x^{2}-8 ki te \frac{5}{2}.

$$5x^{2}-20+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+4 ki te 5.

$$5x^{2}-20+10x+20=2\times 6$$

Tāpirihia te -20 ki te 20, ka 0.

$$5x^{2}+10x=2\times 6$$

Whakareatia te 2 ki te 6, ka 12.

$$5x^{2}+10x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$5x^{2}+10x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 10 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-12\right)}}{2\times 5}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-12\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-10±\sqrt{100-20\left(-12\right)}}{2\times 5}$$

Whakareatia -20 ki te -12.

$$x=\frac{-10±\sqrt{100+240}}{2\times 5}$$

Tāpiri 100 ki te 240.

$$x=\frac{-10±\sqrt{340}}{2\times 5}$$

Tuhia te pūtakerua o te 340.

$$x=\frac{-10±2\sqrt{85}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-10±2\sqrt{85}}{10}$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{85}}{10} ina he tāpiri te ±. Tāpiri -10 ki te 2\sqrt{85}.

$$x=\frac{2\sqrt{85}-10}{10}$$

Whakawehe -10+2\sqrt{85} ki te 10.

$$x=\frac{\sqrt{85}}{5}-1$$

Nā, me whakaoti te whārite x=\frac{-10±2\sqrt{85}}{10} ina he tango te ±. Tango 2\sqrt{85} mai i -10.

$$x=\frac{-2\sqrt{85}-10}{10}$$

Whakawehe -10-2\sqrt{85} ki te 10.

$$x=-\frac{\sqrt{85}}{5}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{85}}{5}-1 x=-\frac{\sqrt{85}}{5}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2,x-2,x^{2}-4.

$$2\left(x-2\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-2.

$$\left(2x-4\right)\left(x+2\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-4 ki te x+2 ka whakakotahi i ngā kupu rite.

$$\left(2x^{2}-8\right)\times \frac{5}{2}+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x^{2}-8 ki te \frac{5}{2}.

$$5x^{2}-20+\left(2x+4\right)\times 5=2\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+4 ki te 5.

$$5x^{2}-20+10x+20=2\times 6$$

Tāpirihia te -20 ki te 20, ka 0.

$$5x^{2}+10x=2\times 6$$

Whakareatia te 2 ki te 6, ka 12.

$$5x^{2}+10x=12$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+10x}{5}=\frac{12}{5}$$

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

$$x^{2}+\frac{10}{5}x=\frac{12}{5}$$

Whakawehe 10 ki te 5.

$$x^{2}+2x=\frac{12}{5}$$

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.

$$x^{2}+2x+1^{2}=\frac{12}{5}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{12}{5}+1$$

Tāpiri \frac{12}{5} ki te 1.

$$x^{2}+2x+1=\frac{17}{5}$$

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}.

$$\left(x+1\right)^{2}=\frac{17}{5}$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{17}{5}}$$

Whakarūnātia.

$$x+1=\frac{\sqrt{85}}{5} x+1=-\frac{\sqrt{85}}{5}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{85}}{5}-1 x=-\frac{\sqrt{85}}{5}-1$$