Whakaoti i te (175-x)x=4000

$$(175-x)x=4000$$

Whakaoti mō x

$$x = \frac{15 \sqrt{65} + 175}{2} \approx 147.966933112$$

$$x = \frac{175 - 15 \sqrt{65}}{2} \approx 27.033066888$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 175-x ki te x.

$$175x-x^{2}=4000$$

Tangohia te 4000 mai i ngā taha e rua.

$$175x-x^{2}-4000=0$$

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.

$$-x^{2}+175x-4000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 175 mō b, me -4000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-175±\sqrt{175^{2}-4\left(-1\right)\left(-4000\right)}}{2\left(-1\right)}$$

Pūrua 175.

$$x=\frac{-175±\sqrt{30625-4\left(-1\right)\left(-4000\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-175±\sqrt{30625+4\left(-4000\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -4000.

$$x=\frac{-175±\sqrt{30625-16000}}{2\left(-1\right)}$$

Tāpiri 30625 ki te -16000.

$$x=\frac{-175±\sqrt{14625}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 14625.

$$x=\frac{-175±15\sqrt{65}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-175±15\sqrt{65}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-175±15\sqrt{65}}{-2} ina he tāpiri te ±. Tāpiri -175 ki te 15\sqrt{65}.

$$x=\frac{15\sqrt{65}-175}{-2}$$

Whakawehe -175+15\sqrt{65} ki te -2.

$$x=\frac{175-15\sqrt{65}}{2}$$

Nā, me whakaoti te whārite x=\frac{-175±15\sqrt{65}}{-2} ina he tango te ±. Tango 15\sqrt{65} mai i -175.

$$x=\frac{-15\sqrt{65}-175}{-2}$$

Whakawehe -175-15\sqrt{65} ki te -2.

$$x=\frac{15\sqrt{65}+175}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{175-15\sqrt{65}}{2} x=\frac{15\sqrt{65}+175}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 175-x ki te x.

$$175x-x^{2}=4000$$

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.

$$-x^{2}+175x=4000$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+175x}{-1}=\frac{4000}{-1}$$

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

$$x^{2}+\frac{175}{-1}x=\frac{4000}{-1}$$

Whakawehe 175 ki te -1.

$$x^{2}-175x=\frac{4000}{-1}$$

Whakawehe 4000 ki te -1.

$$x^{2}-175x=-4000$$

Whakawehea te -175, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{175}{2}. Nā, tāpiria te pūrua o te -\frac{175}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-175x+\left(-\frac{175}{2}\right)^{2}=-4000+\left(-\frac{175}{2}\right)^{2}$$

Pūruatia -\frac{175}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-175x+\frac{30625}{4}=-4000+\frac{30625}{4}$$

Tāpiri -4000 ki te \frac{30625}{4}.

$$x^{2}-175x+\frac{30625}{4}=\frac{14625}{4}$$

Tauwehea x^{2}-175x+\frac{30625}{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}.

$$\left(x-\frac{175}{2}\right)^{2}=\frac{14625}{4}$$

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

$$\sqrt{\left(x-\frac{175}{2}\right)^{2}}=\sqrt{\frac{14625}{4}}$$

Whakarūnātia.

$$x-\frac{175}{2}=\frac{15\sqrt{65}}{2} x-\frac{175}{2}=-\frac{15\sqrt{65}}{2}$$

Me tāpiri \frac{175}{2} ki ngā taha e rua o te whārite.

$$x=\frac{15\sqrt{65}+175}{2} x=\frac{175-15\sqrt{65}}{2}$$