Whakaoti i te 70/x+35+70/x-35=40

$$\frac{ 70 }{ x+35 } + \frac{ 70 }{ x-35 } =40$$

Whakaoti mō x

$$x = \frac{7 \sqrt{401} + 7}{4} \approx 36.79372269$$

$$x=\frac{7-7\sqrt{401}}{4}\approx -33.29372269$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -35,35 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-35\right)\left(x+35\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+35,x-35.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-35 ki te 70.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+35 ki te 70.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

Pahekotia te 70x me 70x, ka 140x.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

Tāpirihia te -2450 ki te 2450, ka 0.

$$140x=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 40 ki te x-35.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 40x-1400 ki te x+35 ka whakakotahi i ngā kupu rite.

$$140x=40x^{2}-49000$$

Tangohia te 40x^{2} mai i ngā taha e rua.

$$140x-40x^{2}=-49000$$

Me tāpiri te 49000 ki ngā taha e rua.

$$140x-40x^{2}+49000=0$$

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.

$$-40x^{2}+140x+49000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -40 mō a, 140 mō b, me 49000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

Pūrua 140.

$$x=\frac{-140±\sqrt{19600-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

Whakareatia -4 ki te -40.

$$x=\frac{-140±\sqrt{19600+160\times 49000}}{2\left(-40\right)}$$

Whakareatia 160 ki te 49000.

$$x=\frac{-140±\sqrt{19600+7840000}}{2\left(-40\right)}$$

Tāpiri 19600 ki te 7840000.

$$x=\frac{-140±\sqrt{7859600}}{2\left(-40\right)}$$

Tuhia te pūtakerua o te 7859600.

$$x=\frac{-140±140\sqrt{401}}{2\left(-40\right)}$$

Whakareatia 2 ki te -40.

$$x=\frac{-140±140\sqrt{401}}{-80}$$

Nā, me whakaoti te whārite x=\frac{-140±140\sqrt{401}}{-80} ina he tāpiri te ±. Tāpiri -140 ki te 140\sqrt{401}.

$$x=\frac{140\sqrt{401}-140}{-80}$$

Whakawehe -140+140\sqrt{401} ki te -80.

$$x=\frac{7-7\sqrt{401}}{4}$$

Nā, me whakaoti te whārite x=\frac{-140±140\sqrt{401}}{-80} ina he tango te ±. Tango 140\sqrt{401} mai i -140.

$$x=\frac{-140\sqrt{401}-140}{-80}$$

Whakawehe -140-140\sqrt{401} ki te -80.

$$x=\frac{7\sqrt{401}+7}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{7-7\sqrt{401}}{4} x=\frac{7\sqrt{401}+7}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -35,35 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-35\right)\left(x+35\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+35,x-35.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-35 ki te 70.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+35 ki te 70.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

Pahekotia te 70x me 70x, ka 140x.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

Tāpirihia te -2450 ki te 2450, ka 0.

$$140x=40\left(x-35\right)\left(x+35\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 40 ki te x-35.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 40x-1400 ki te x+35 ka whakakotahi i ngā kupu rite.

$$140x=40x^{2}-49000$$

Tangohia te 40x^{2} mai i ngā taha e rua.

$$140x-40x^{2}=-49000$$

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.

$$-40x^{2}+140x=-49000$$

Whakawehea ngā taha e rua ki te -40.

$$\frac{-40x^{2}+140x}{-40}=-\frac{49000}{-40}$$

Mā te whakawehe ki te -40 ka wetekia te whakareanga ki te -40.

$$x^{2}+\frac{140}{-40}x=-\frac{49000}{-40}$$

Whakahekea te hautanga \frac{140}{-40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x^{2}-\frac{7}{2}x=-\frac{49000}{-40}$$

Whakawehe -49000 ki te -40.

$$x^{2}-\frac{7}{2}x=1225$$

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=1225+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=1225+\frac{49}{16}$$

Tāpiri 1225 ki te \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{19649}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{19649}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{19649}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{7\sqrt{401}}{4} x-\frac{7}{4}=-\frac{7\sqrt{401}}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$x=\frac{7\sqrt{401}+7}{4} x=\frac{7-7\sqrt{401}}{4}$$