Whakaoti i te (90-30x)(20x)=50

$$(90-30x)(20x) = 50$$

Whakaoti mō x

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 2.971960144$$

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 0.028039856$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 90-30x ki te 20.

$$\left(1800-600x\right)x=50$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1800-600x ki te x.

$$1800x-600x^{2}=50$$

Tangohia te 50 mai i ngā taha e rua.

$$1800x-600x^{2}-50=0$$

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.

$$-600x^{2}+1800x-50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -600 mō a, 1800 mō b, me -50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1800±\sqrt{1800^{2}-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

Pūrua 1800.

$$x=\frac{-1800±\sqrt{3240000-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

Whakareatia -4 ki te -600.

$$x=\frac{-1800±\sqrt{3240000+2400\left(-50\right)}}{2\left(-600\right)}$$

Whakareatia 2400 ki te -50.

$$x=\frac{-1800±\sqrt{3240000-120000}}{2\left(-600\right)}$$

Tāpiri 3240000 ki te -120000.

$$x=\frac{-1800±\sqrt{3120000}}{2\left(-600\right)}$$

Tuhia te pūtakerua o te 3120000.

$$x=\frac{-1800±200\sqrt{78}}{2\left(-600\right)}$$

Whakareatia 2 ki te -600.

$$x=\frac{-1800±200\sqrt{78}}{-1200}$$

Nā, me whakaoti te whārite x=\frac{-1800±200\sqrt{78}}{-1200} ina he tāpiri te ±. Tāpiri -1800 ki te 200\sqrt{78}.

$$x=\frac{200\sqrt{78}-1800}{-1200}$$

Whakawehe -1800+200\sqrt{78} ki te -1200.

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{-1800±200\sqrt{78}}{-1200} ina he tango te ±. Tango 200\sqrt{78} mai i -1800.

$$x=\frac{-200\sqrt{78}-1800}{-1200}$$

Whakawehe -1800-200\sqrt{78} ki te -1200.

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2} x=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 90-30x ki te 20.

$$\left(1800-600x\right)x=50$$

Whakamahia te āhuatanga tohatoha hei whakarea te 1800-600x ki te x.

$$1800x-600x^{2}=50$$

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.

$$-600x^{2}+1800x=50$$

Whakawehea ngā taha e rua ki te -600.

$$\frac{-600x^{2}+1800x}{-600}=\frac{50}{-600}$$

Mā te whakawehe ki te -600 ka wetekia te whakareanga ki te -600.

$$x^{2}+\frac{1800}{-600}x=\frac{50}{-600}$$

Whakawehe 1800 ki te -600.

$$x^{2}-3x=\frac{50}{-600}$$

Whakahekea te hautanga \frac{50}{-600} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 50.

$$x^{2}-3x=-\frac{1}{12}$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-\frac{1}{12}+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=-\frac{1}{12}+\frac{9}{4}$$

Tāpiri -\frac{1}{12} ki te \frac{9}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-3x+\frac{9}{4}=\frac{13}{6}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{13}{6}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{13}{6}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{78}}{6} x-\frac{3}{2}=-\frac{\sqrt{78}}{6}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2} x=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$