Whakaoti i te 72/x-72/x+10=36

$$\frac { 72 } { x } - \frac { 72 } { x + 10 } = 36$$

Whakaoti mō x

$$x=3\sqrt{5}-5\approx 1.708203932$$

$$x=-3\sqrt{5}-5\approx -11.708203932$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+10.

$$\left(x+10\right)\times 72-x\times 72=36x\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+10 ki te 72.

$$72x+720-x\times 72=36x\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 36x ki te x+10.

$$72x+720-x\times 72=36x^{2}+360x$$

Tangohia te 36x^{2} mai i ngā taha e rua.

$$72x+720-x\times 72-36x^{2}=360x$$

Tangohia te 360x mai i ngā taha e rua.

$$72x+720-x\times 72-36x^{2}-360x=0$$

Pahekotia te 72x me -360x, ka -288x.

$$-288x+720-x\times 72-36x^{2}=0$$

Whakareatia te -1 ki te 72, ka -72.

$$-288x+720-72x-36x^{2}=0$$

Pahekotia te -288x me -72x, ka -360x.

$$-360x+720-36x^{2}=0$$

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.

$$-36x^{2}-360x+720=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -36 mō a, -360 mō b, me 720 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\left(-36\right)\times 720}}{2\left(-36\right)}$$

Pūrua -360.

$$x=\frac{-\left(-360\right)±\sqrt{129600-4\left(-36\right)\times 720}}{2\left(-36\right)}$$

Whakareatia -4 ki te -36.

$$x=\frac{-\left(-360\right)±\sqrt{129600+144\times 720}}{2\left(-36\right)}$$

Whakareatia 144 ki te 720.

$$x=\frac{-\left(-360\right)±\sqrt{129600+103680}}{2\left(-36\right)}$$

Tāpiri 129600 ki te 103680.

$$x=\frac{-\left(-360\right)±\sqrt{233280}}{2\left(-36\right)}$$

Tuhia te pūtakerua o te 233280.

$$x=\frac{-\left(-360\right)±216\sqrt{5}}{2\left(-36\right)}$$

Ko te tauaro o -360 ko 360.

$$x=\frac{360±216\sqrt{5}}{2\left(-36\right)}$$

Whakareatia 2 ki te -36.

$$x=\frac{360±216\sqrt{5}}{-72}$$

Nā, me whakaoti te whārite x=\frac{360±216\sqrt{5}}{-72} ina he tāpiri te ±. Tāpiri 360 ki te 216\sqrt{5}.

$$x=\frac{216\sqrt{5}+360}{-72}$$

Whakawehe 360+216\sqrt{5} ki te -72.

$$x=-3\sqrt{5}-5$$

Nā, me whakaoti te whārite x=\frac{360±216\sqrt{5}}{-72} ina he tango te ±. Tango 216\sqrt{5} mai i 360.

$$x=\frac{360-216\sqrt{5}}{-72}$$

Whakawehe 360-216\sqrt{5} ki te -72.

$$x=3\sqrt{5}-5$$

Kua oti te whārite te whakatau.

$$x=-3\sqrt{5}-5 x=3\sqrt{5}-5$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -10,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+10\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+10.

$$\left(x+10\right)\times 72-x\times 72=36x\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+10 ki te 72.

$$72x+720-x\times 72=36x\left(x+10\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 36x ki te x+10.

$$72x+720-x\times 72=36x^{2}+360x$$

Tangohia te 36x^{2} mai i ngā taha e rua.

$$72x+720-x\times 72-36x^{2}=360x$$

Tangohia te 360x mai i ngā taha e rua.

$$72x+720-x\times 72-36x^{2}-360x=0$$

Pahekotia te 72x me -360x, ka -288x.

$$-288x+720-x\times 72-36x^{2}=0$$

Tangohia te 720 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-288x-x\times 72-36x^{2}=-720$$

Whakareatia te -1 ki te 72, ka -72.

$$-288x-72x-36x^{2}=-720$$

Pahekotia te -288x me -72x, ka -360x.

$$-360x-36x^{2}=-720$$

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.

$$-36x^{2}-360x=-720$$

Whakawehea ngā taha e rua ki te -36.

$$\frac{-36x^{2}-360x}{-36}=-\frac{720}{-36}$$

Mā te whakawehe ki te -36 ka wetekia te whakareanga ki te -36.

$$x^{2}+\left(-\frac{360}{-36}\right)x=-\frac{720}{-36}$$

Whakawehe -360 ki te -36.

$$x^{2}+10x=-\frac{720}{-36}$$

Whakawehe -720 ki te -36.

$$x^{2}+10x=20$$

Whakawehea te 10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 5. Nā, tāpiria te pūrua o te 5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+10x+5^{2}=20+5^{2}$$

Pūrua 5.

$$x^{2}+10x+25=20+25$$

Tāpiri 20 ki te 25.

$$x^{2}+10x+25=45$$

Tauwehea x^{2}+10x+25. 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}.

$$\left(x+5\right)^{2}=45$$

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

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{45}$$

Whakarūnātia.

$$x+5=3\sqrt{5} x+5=-3\sqrt{5}$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$x=3\sqrt{5}-5 x=-3\sqrt{5}-5$$