Whakaoti i te 1/x+1/x-7=1/12

$$\frac { 1 } { x } + \frac { 1 } { x - 7 } = \frac { 1 } { 12 }$$

Whakaoti mō x

$$x=3$$

$$x=28$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,7 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x\left(x-7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

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

$$24x-84=x\left(x-7\right)$$

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

$$24x-84=x^{2}-7x$$

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

$$24x-84-x^{2}=-7x$$

Me tāpiri te 7x ki ngā taha e rua.

$$24x-84-x^{2}+7x=0$$

Pahekotia te 24x me 7x, ka 31x.

$$31x-84-x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+31x-84=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-84. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=31 ab=-\left(-84\right)=84$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 84.

$$1,84 2,42 3,28 4,21 6,14 7,12$$

Tātaihia te tapeke mō ia takirua.

$$1+84=85 2+42=44 3+28=31 4+21=25 6+14=20 7+12=19$$

Ko te otinga te takirua ka hoatu i te tapeke 31.

$$a=28 b=3$$

Tuhia anō te -x^{2}+31x-84 hei \left(-x^{2}+28x\right)+\left(3x-84\right).

$$\left(-x^{2}+28x\right)+\left(3x-84\right)$$

Tauwehea te -x i te tuatahi me te 3 i te rōpū tuarua.

$$-x\left(x-28\right)+3\left(x-28\right)$$

Whakatauwehea atu te kīanga pātahi x-28 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-28\right)\left(-x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-28=0 me te -x+3=0.

$$x=28 x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,7 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x\left(x-7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

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

$$24x-84=x\left(x-7\right)$$

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

$$24x-84=x^{2}-7x$$

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

$$24x-84-x^{2}=-7x$$

Me tāpiri te 7x ki ngā taha e rua.

$$24x-84-x^{2}+7x=0$$

Pahekotia te 24x me 7x, ka 31x.

$$31x-84-x^{2}=0$$

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.

$$-x^{2}+31x-84=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 31 mō b, me -84 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Pūrua 31.

$$x=\frac{-31±\sqrt{961-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-31±\sqrt{961+4\left(-84\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -84.

$$x=\frac{-31±\sqrt{961-336}}{2\left(-1\right)}$$

Tāpiri 961 ki te -336.

$$x=\frac{-31±\sqrt{625}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-31±25}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-31±25}{-2}$$

Nā, me whakaoti te whārite x=\frac{-31±25}{-2} ina he tāpiri te ±. Tāpiri -31 ki te 25.

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

Whakawehe -6 ki te -2.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{-31±25}{-2} ina he tango te ±. Tango 25 mai i -31.

$$x=-\frac{56}{-2}$$

Whakawehe -56 ki te -2.

$$x=28$$

Kua oti te whārite te whakatau.

$$x=3 x=28$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,7 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 12x\left(x-7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-7,12.

$$12x-84+12x=x\left(x-7\right)$$

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

$$24x-84=x\left(x-7\right)$$

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

$$24x-84=x^{2}-7x$$

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

$$24x-84-x^{2}=-7x$$

Me tāpiri te 7x ki ngā taha e rua.

$$24x-84-x^{2}+7x=0$$

Pahekotia te 24x me 7x, ka 31x.

$$31x-84-x^{2}=0$$

Me tāpiri te 84 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$31x-x^{2}=84$$

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.

$$-x^{2}+31x=84$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+31x}{-1}=\frac{84}{-1}$$

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

$$x^{2}+\frac{31}{-1}x=\frac{84}{-1}$$

Whakawehe 31 ki te -1.

$$x^{2}-31x=\frac{84}{-1}$$

Whakawehe 84 ki te -1.

$$x^{2}-31x=-84$$

Whakawehea te -31, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{31}{2}. Nā, tāpiria te pūrua o te -\frac{31}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-31x+\left(-\frac{31}{2}\right)^{2}=-84+\left(-\frac{31}{2}\right)^{2}$$

Pūruatia -\frac{31}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-31x+\frac{961}{4}=-84+\frac{961}{4}$$

Tāpiri -84 ki te \frac{961}{4}.

$$x^{2}-31x+\frac{961}{4}=\frac{625}{4}$$

Tauwehea x^{2}-31x+\frac{961}{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}.

$$\left(x-\frac{31}{2}\right)^{2}=\frac{625}{4}$$

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

$$\sqrt{\left(x-\frac{31}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Whakarūnātia.

$$x-\frac{31}{2}=\frac{25}{2} x-\frac{31}{2}=-\frac{25}{2}$$

Me tāpiri \frac{31}{2} ki ngā taha e rua o te whārite.

$$x=28 x=3$$