Whakaoti i te 2400/x-50/x+15=9

$$\frac { 2400 } { x } - \frac { 50 } { x + 15 } = 9$$

Whakaoti mō x

$$x = \frac{5 \sqrt{248089} + 2215}{18} \approx 261.412592793$$

$$x=\frac{2215-5\sqrt{248089}}{18}\approx -15.301481682$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -15,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+15\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+15.

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+15 ki te 2400.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9x ki te x+15.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Tangohia te 135x mai i ngā taha e rua.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

Pahekotia te 2400x me -135x, ka 2265x.

$$2265x+36000-x\times 50-9x^{2}=0$$

Whakareatia te -1 ki te 50, ka -50.

$$2265x+36000-50x-9x^{2}=0$$

Pahekotia te 2265x me -50x, ka 2215x.

$$2215x+36000-9x^{2}=0$$

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.

$$-9x^{2}+2215x+36000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, 2215 mō b, me 36000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2215±\sqrt{2215^{2}-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

Pūrua 2215.

$$x=\frac{-2215±\sqrt{4906225-4\left(-9\right)\times 36000}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-2215±\sqrt{4906225+36\times 36000}}{2\left(-9\right)}$$

Whakareatia 36 ki te 36000.

$$x=\frac{-2215±\sqrt{4906225+1296000}}{2\left(-9\right)}$$

Tāpiri 4906225 ki te 1296000.

$$x=\frac{-2215±\sqrt{6202225}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 6202225.

$$x=\frac{-2215±5\sqrt{248089}}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{-2215±5\sqrt{248089}}{-18}$$

Nā, me whakaoti te whārite x=\frac{-2215±5\sqrt{248089}}{-18} ina he tāpiri te ±. Tāpiri -2215 ki te 5\sqrt{248089}.

$$x=\frac{5\sqrt{248089}-2215}{-18}$$

Whakawehe -2215+5\sqrt{248089} ki te -18.

$$x=\frac{2215-5\sqrt{248089}}{18}$$

Nā, me whakaoti te whārite x=\frac{-2215±5\sqrt{248089}}{-18} ina he tango te ±. Tango 5\sqrt{248089} mai i -2215.

$$x=\frac{-5\sqrt{248089}-2215}{-18}$$

Whakawehe -2215-5\sqrt{248089} ki te -18.

$$x=\frac{5\sqrt{248089}+2215}{18}$$

Kua oti te whārite te whakatau.

$$x=\frac{2215-5\sqrt{248089}}{18} x=\frac{5\sqrt{248089}+2215}{18}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -15,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+15\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+15.

$$\left(x+15\right)\times 2400-x\times 50=9x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+15 ki te 2400.

$$2400x+36000-x\times 50=9x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 9x ki te x+15.

$$2400x+36000-x\times 50=9x^{2}+135x$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$2400x+36000-x\times 50-9x^{2}=135x$$

Tangohia te 135x mai i ngā taha e rua.

$$2400x+36000-x\times 50-9x^{2}-135x=0$$

Pahekotia te 2400x me -135x, ka 2265x.

$$2265x+36000-x\times 50-9x^{2}=0$$

Tangohia te 36000 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2265x-x\times 50-9x^{2}=-36000$$

Whakareatia te -1 ki te 50, ka -50.

$$2265x-50x-9x^{2}=-36000$$

Pahekotia te 2265x me -50x, ka 2215x.

$$2215x-9x^{2}=-36000$$

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.

$$-9x^{2}+2215x=-36000$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}+2215x}{-9}=-\frac{36000}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$x^{2}+\frac{2215}{-9}x=-\frac{36000}{-9}$$

Whakawehe 2215 ki te -9.

$$x^{2}-\frac{2215}{9}x=-\frac{36000}{-9}$$

Whakawehe -36000 ki te -9.

$$x^{2}-\frac{2215}{9}x=4000$$

Whakawehea te -\frac{2215}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2215}{18}. Nā, tāpiria te pūrua o te -\frac{2215}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2215}{9}x+\left(-\frac{2215}{18}\right)^{2}=4000+\left(-\frac{2215}{18}\right)^{2}$$

Pūruatia -\frac{2215}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=4000+\frac{4906225}{324}$$

Tāpiri 4000 ki te \frac{4906225}{324}.

$$x^{2}-\frac{2215}{9}x+\frac{4906225}{324}=\frac{6202225}{324}$$

Tauwehea x^{2}-\frac{2215}{9}x+\frac{4906225}{324}. 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}.

$$\left(x-\frac{2215}{18}\right)^{2}=\frac{6202225}{324}$$

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

$$\sqrt{\left(x-\frac{2215}{18}\right)^{2}}=\sqrt{\frac{6202225}{324}}$$

Whakarūnātia.

$$x-\frac{2215}{18}=\frac{5\sqrt{248089}}{18} x-\frac{2215}{18}=-\frac{5\sqrt{248089}}{18}$$

Me tāpiri \frac{2215}{18} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{248089}+2215}{18} x=\frac{2215-5\sqrt{248089}}{18}$$