Whakaoti i te 1/x+1+1/x+2=13/12

$$\frac { 1 } { x + 1 } + \frac { 1 } { x + 2 } = \frac { 13 } { 12 }$$

Whakaoti mō x

$$x=\frac{\sqrt{745}-15}{26}\approx 0.47287262$$

$$x=\frac{-\sqrt{745}-15}{26}\approx -1.626718774$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x+2,12.

$$12x+24+12x+12=13\left(x+1\right)\left(x+2\right)$$

Pahekotia te 12x me 12x, ka 24x.

$$24x+24+12=13\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 24 ki te 12, ka 36.

$$24x+36=13\left(x+1\right)\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13 ki te x+1.

$$24x+36=\left(13x+13\right)\left(x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 13x+13 ki te x+2 ka whakakotahi i ngā kupu rite.

$$24x+36=13x^{2}+39x+26$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$24x+36-13x^{2}=39x+26$$

Tangohia te 39x mai i ngā taha e rua.

$$24x+36-13x^{2}-39x=26$$

Pahekotia te 24x me -39x, ka -15x.

$$-15x+36-13x^{2}=26$$

Tangohia te 26 mai i ngā taha e rua.

$$-15x+36-13x^{2}-26=0$$

Tangohia te 26 i te 36, ka 10.

$$-15x+10-13x^{2}=0$$

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.

$$-13x^{2}-15x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -13 mō a, -15 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Whakareatia -4 ki te -13.

$$x=\frac{-\left(-15\right)±\sqrt{225+52\times 10}}{2\left(-13\right)}$$

Whakareatia 52 ki te 10.

$$x=\frac{-\left(-15\right)±\sqrt{225+520}}{2\left(-13\right)}$$

Tāpiri 225 ki te 520.

$$x=\frac{-\left(-15\right)±\sqrt{745}}{2\left(-13\right)}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±\sqrt{745}}{2\left(-13\right)}$$

Whakareatia 2 ki te -13.

$$x=\frac{15±\sqrt{745}}{-26}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{745}}{-26} ina he tāpiri te ±. Tāpiri 15 ki te \sqrt{745}.

$$x=\frac{\sqrt{745}+15}{-26}$$

Whakawehe 15+\sqrt{745} ki te -26.

$$x=\frac{-\sqrt{745}-15}{26}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{745}}{-26} ina he tango te ±. Tango \sqrt{745} mai i 15.

$$x=\frac{15-\sqrt{745}}{-26}$$

Whakawehe 15-\sqrt{745} ki te -26.

$$x=\frac{\sqrt{745}-15}{26}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{745}-15}{26} x=\frac{\sqrt{745}-15}{26}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x+2,12.

$$12x+24+12x+12=13\left(x+1\right)\left(x+2\right)$$

Pahekotia te 12x me 12x, ka 24x.

$$24x+24+12=13\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 24 ki te 12, ka 36.

$$24x+36=13\left(x+1\right)\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13 ki te x+1.

$$24x+36=\left(13x+13\right)\left(x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 13x+13 ki te x+2 ka whakakotahi i ngā kupu rite.

$$24x+36=13x^{2}+39x+26$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$24x+36-13x^{2}=39x+26$$

Tangohia te 39x mai i ngā taha e rua.

$$24x+36-13x^{2}-39x=26$$

Pahekotia te 24x me -39x, ka -15x.

$$-15x+36-13x^{2}=26$$

Tangohia te 36 mai i ngā taha e rua.

$$-15x-13x^{2}=26-36$$

Tangohia te 36 i te 26, ka -10.

$$-15x-13x^{2}=-10$$

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.

$$-13x^{2}-15x=-10$$

Whakawehea ngā taha e rua ki te -13.

$$\frac{-13x^{2}-15x}{-13}=-\frac{10}{-13}$$

Mā te whakawehe ki te -13 ka wetekia te whakareanga ki te -13.

$$x^{2}+\left(-\frac{15}{-13}\right)x=-\frac{10}{-13}$$

Whakawehe -15 ki te -13.

$$x^{2}+\frac{15}{13}x=-\frac{10}{-13}$$

Whakawehe -10 ki te -13.

$$x^{2}+\frac{15}{13}x=\frac{10}{13}$$

Whakawehea te \frac{15}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{26}. Nā, tāpiria te pūrua o te \frac{15}{26} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{15}{13}x+\left(\frac{15}{26}\right)^{2}=\frac{10}{13}+\left(\frac{15}{26}\right)^{2}$$

Pūruatia \frac{15}{26} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{15}{13}x+\frac{225}{676}=\frac{10}{13}+\frac{225}{676}$$

Tāpiri \frac{10}{13} ki te \frac{225}{676} 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.

$$x^{2}+\frac{15}{13}x+\frac{225}{676}=\frac{745}{676}$$

Tauwehea x^{2}+\frac{15}{13}x+\frac{225}{676}. 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}.

$$\left(x+\frac{15}{26}\right)^{2}=\frac{745}{676}$$

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

$$\sqrt{\left(x+\frac{15}{26}\right)^{2}}=\sqrt{\frac{745}{676}}$$

Whakarūnātia.

$$x+\frac{15}{26}=\frac{\sqrt{745}}{26} x+\frac{15}{26}=-\frac{\sqrt{745}}{26}$$

Me tango \frac{15}{26} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{745}-15}{26} x=\frac{-\sqrt{745}-15}{26}$$