Whakaoti i te 20/x-5=60/x+5+1

$$\frac{ 20 }{ x-5 } = \frac{ 60 }{ x+5 } +1$$

Whakaoti mō x

$$x=5\sqrt{33}-20\approx 8.722813233$$

$$x=-5\sqrt{33}-20\approx -48.722813233$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+5.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+5 ki te 20.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 60.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Whakaarohia te \left(x-5\right)\left(x+5\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 5.

$$20x+100=60x-300+x^{2}-25$$

Tangohia te 25 i te -300, ka -325.

$$20x+100=60x-325+x^{2}$$

Tangohia te 60x mai i ngā taha e rua.

$$20x+100-60x=-325+x^{2}$$

Pahekotia te 20x me -60x, ka -40x.

$$-40x+100=-325+x^{2}$$

Tangohia te -325 mai i ngā taha e rua.

$$-40x+100-\left(-325\right)=x^{2}$$

Ko te tauaro o -325 ko 325.

$$-40x+100+325=x^{2}$$

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

$$-40x+100+325-x^{2}=0$$

Tāpirihia te 100 ki te 325, ka 425.

$$-40x+425-x^{2}=0$$

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.

$$-x^{2}-40x+425=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -40 mō b, me 425 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

Pūrua -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-40\right)±\sqrt{1600+4\times 425}}{2\left(-1\right)}$$

Whakareatia 4 ki te 425.

$$x=\frac{-\left(-40\right)±\sqrt{1600+1700}}{2\left(-1\right)}$$

Tāpiri 1600 ki te 1700.

$$x=\frac{-\left(-40\right)±\sqrt{3300}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 3300.

$$x=\frac{-\left(-40\right)±10\sqrt{33}}{2\left(-1\right)}$$

Ko te tauaro o -40 ko 40.

$$x=\frac{40±10\sqrt{33}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{40±10\sqrt{33}}{-2}$$

Nā, me whakaoti te whārite x=\frac{40±10\sqrt{33}}{-2} ina he tāpiri te ±. Tāpiri 40 ki te 10\sqrt{33}.

$$x=\frac{10\sqrt{33}+40}{-2}$$

Whakawehe 40+10\sqrt{33} ki te -2.

$$x=-5\sqrt{33}-20$$

Nā, me whakaoti te whārite x=\frac{40±10\sqrt{33}}{-2} ina he tango te ±. Tango 10\sqrt{33} mai i 40.

$$x=\frac{40-10\sqrt{33}}{-2}$$

Whakawehe 40-10\sqrt{33} ki te -2.

$$x=5\sqrt{33}-20$$

Kua oti te whārite te whakatau.

$$x=-5\sqrt{33}-20 x=5\sqrt{33}-20$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+5.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+5 ki te 20.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 60.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Whakaarohia te \left(x-5\right)\left(x+5\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 5.

$$20x+100=60x-300+x^{2}-25$$

Tangohia te 25 i te -300, ka -325.

$$20x+100=60x-325+x^{2}$$

Tangohia te 60x mai i ngā taha e rua.

$$20x+100-60x=-325+x^{2}$$

Pahekotia te 20x me -60x, ka -40x.

$$-40x+100=-325+x^{2}$$

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

$$-40x+100-x^{2}=-325$$

Tangohia te 100 mai i ngā taha e rua.

$$-40x-x^{2}=-325-100$$

Tangohia te 100 i te -325, ka -425.

$$-40x-x^{2}=-425$$

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.

$$-x^{2}-40x=-425$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-40x}{-1}=-\frac{425}{-1}$$

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

$$x^{2}+\left(-\frac{40}{-1}\right)x=-\frac{425}{-1}$$

Whakawehe -40 ki te -1.

$$x^{2}+40x=-\frac{425}{-1}$$

Whakawehe -425 ki te -1.

$$x^{2}+40x=425$$

Whakawehea te 40, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 20. Nā, tāpiria te pūrua o te 20 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+40x+20^{2}=425+20^{2}$$

Pūrua 20.

$$x^{2}+40x+400=425+400$$

Tāpiri 425 ki te 400.

$$x^{2}+40x+400=825$$

Tauwehea te x^{2}+40x+400. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+20\right)^{2}=825$$

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

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{825}$$

Whakarūnātia.

$$x+20=5\sqrt{33} x+20=-5\sqrt{33}$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$x=5\sqrt{33}-20 x=-5\sqrt{33}-20$$