Whakaoti i te 8/x-3/x+4=5

$$\frac{ 8 }{ x } - \frac{ 3 }{ x+4 } =5$$

Whakaoti mō x

$$x=\frac{\sqrt{865}}{10}-\frac{3}{2}\approx 1.441088234$$

$$x=-\frac{\sqrt{865}}{10}-\frac{3}{2}\approx -4.441088234$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+4\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+4.

$$\left(x+4\right)\times 8-x\times 3=5x\left(x+4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+4 ki te 8.

$$8x+32-x\times 3=5x\left(x+4\right)$$

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

$$8x+32-x\times 3=5x^{2}+20x$$

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

$$8x+32-x\times 3-5x^{2}=20x$$

Tangohia te 20x mai i ngā taha e rua.

$$8x+32-x\times 3-5x^{2}-20x=0$$

Pahekotia te 8x me -20x, ka -12x.

$$-12x+32-x\times 3-5x^{2}=0$$

Whakareatia te -1 ki te 3, ka -3.

$$-12x+32-3x-5x^{2}=0$$

Pahekotia te -12x me -3x, ka -15x.

$$-15x+32-5x^{2}=0$$

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.

$$-5x^{2}-15x+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, -15 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-5\right)\times 32}}{2\left(-5\right)}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-5\right)\times 32}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-\left(-15\right)±\sqrt{225+20\times 32}}{2\left(-5\right)}$$

Whakareatia 20 ki te 32.

$$x=\frac{-\left(-15\right)±\sqrt{225+640}}{2\left(-5\right)}$$

Tāpiri 225 ki te 640.

$$x=\frac{-\left(-15\right)±\sqrt{865}}{2\left(-5\right)}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±\sqrt{865}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{15±\sqrt{865}}{-10}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{865}}{-10} ina he tāpiri te ±. Tāpiri 15 ki te \sqrt{865}.

$$x=\frac{\sqrt{865}+15}{-10}$$

Whakawehe 15+\sqrt{865} ki te -10.

$$x=-\frac{\sqrt{865}}{10}-\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{865}}{-10} ina he tango te ±. Tango \sqrt{865} mai i 15.

$$x=\frac{15-\sqrt{865}}{-10}$$

Whakawehe 15-\sqrt{865} ki te -10.

$$x=\frac{\sqrt{865}}{10}-\frac{3}{2}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{865}}{10}-\frac{3}{2} x=\frac{\sqrt{865}}{10}-\frac{3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+4\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+4.

$$\left(x+4\right)\times 8-x\times 3=5x\left(x+4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+4 ki te 8.

$$8x+32-x\times 3=5x\left(x+4\right)$$

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

$$8x+32-x\times 3=5x^{2}+20x$$

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

$$8x+32-x\times 3-5x^{2}=20x$$

Tangohia te 20x mai i ngā taha e rua.

$$8x+32-x\times 3-5x^{2}-20x=0$$

Pahekotia te 8x me -20x, ka -12x.

$$-12x+32-x\times 3-5x^{2}=0$$

Tangohia te 32 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-12x-x\times 3-5x^{2}=-32$$

Whakareatia te -1 ki te 3, ka -3.

$$-12x-3x-5x^{2}=-32$$

Pahekotia te -12x me -3x, ka -15x.

$$-15x-5x^{2}=-32$$

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.

$$-5x^{2}-15x=-32$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5x^{2}-15x}{-5}=-\frac{32}{-5}$$

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

$$x^{2}+\left(-\frac{15}{-5}\right)x=-\frac{32}{-5}$$

Whakawehe -15 ki te -5.

$$x^{2}+3x=-\frac{32}{-5}$$

Whakawehe -32 ki te -5.

$$x^{2}+3x=\frac{32}{5}$$

Whakawehea te 3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2}. Nā, tāpiria te pūrua o te \frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{32}{5}+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=\frac{32}{5}+\frac{9}{4}$$

Tāpiri \frac{32}{5} ki te \frac{9}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+3x+\frac{9}{4}=\frac{173}{20}$$

Tauwehea x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{173}{20}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{173}{20}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{\sqrt{865}}{10} x+\frac{3}{2}=-\frac{\sqrt{865}}{10}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{865}}{10}-\frac{3}{2} x=-\frac{\sqrt{865}}{10}-\frac{3}{2}$$