Whakaoti i te (20-x)(100+20x)=2240

$$(20-x)(100+20x)=2240$$

Whakaoti mō x

$$x = \frac{\sqrt{177} + 15}{2} \approx 14.152067348$$

$$x=\frac{15-\sqrt{177}}{2}\approx 0.847932652$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 100+20x ka whakakotahi i ngā kupu rite.

$$2000+300x-20x^{2}=2240$$

Tangohia te 2240 mai i ngā taha e rua.

$$2000+300x-20x^{2}-2240=0$$

Tangohia te 2240 i te 2000, ka -240.

$$-240+300x-20x^{2}=0$$

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.

$$-20x^{2}+300x-240=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -20 mō a, 300 mō b, me -240 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

Pūrua 300.

$$x=\frac{-300±\sqrt{90000-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

Whakareatia -4 ki te -20.

$$x=\frac{-300±\sqrt{90000+80\left(-240\right)}}{2\left(-20\right)}$$

Whakareatia 80 ki te -240.

$$x=\frac{-300±\sqrt{90000-19200}}{2\left(-20\right)}$$

Tāpiri 90000 ki te -19200.

$$x=\frac{-300±\sqrt{70800}}{2\left(-20\right)}$$

Tuhia te pūtakerua o te 70800.

$$x=\frac{-300±20\sqrt{177}}{2\left(-20\right)}$$

Whakareatia 2 ki te -20.

$$x=\frac{-300±20\sqrt{177}}{-40}$$

Nā, me whakaoti te whārite x=\frac{-300±20\sqrt{177}}{-40} ina he tāpiri te ±. Tāpiri -300 ki te 20\sqrt{177}.

$$x=\frac{20\sqrt{177}-300}{-40}$$

Whakawehe -300+20\sqrt{177} ki te -40.

$$x=\frac{15-\sqrt{177}}{2}$$

Nā, me whakaoti te whārite x=\frac{-300±20\sqrt{177}}{-40} ina he tango te ±. Tango 20\sqrt{177} mai i -300.

$$x=\frac{-20\sqrt{177}-300}{-40}$$

Whakawehe -300-20\sqrt{177} ki te -40.

$$x=\frac{\sqrt{177}+15}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{15-\sqrt{177}}{2} x=\frac{\sqrt{177}+15}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-x ki te 100+20x ka whakakotahi i ngā kupu rite.

$$2000+300x-20x^{2}=2240$$

Tangohia te 2000 mai i ngā taha e rua.

$$300x-20x^{2}=2240-2000$$

Tangohia te 2000 i te 2240, ka 240.

$$300x-20x^{2}=240$$

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.

$$-20x^{2}+300x=240$$

Whakawehea ngā taha e rua ki te -20.

$$\frac{-20x^{2}+300x}{-20}=\frac{240}{-20}$$

Mā te whakawehe ki te -20 ka wetekia te whakareanga ki te -20.

$$x^{2}+\frac{300}{-20}x=\frac{240}{-20}$$

Whakawehe 300 ki te -20.

$$x^{2}-15x=\frac{240}{-20}$$

Whakawehe 240 ki te -20.

$$x^{2}-15x=-12$$

Whakawehea te -15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{2}. Nā, tāpiria te pūrua o te -\frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-12+\left(-\frac{15}{2}\right)^{2}$$

Pūruatia -\frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-15x+\frac{225}{4}=-12+\frac{225}{4}$$

Tāpiri -12 ki te \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{177}{4}$$

Tauwehea x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{177}{4}$$

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

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{177}{4}}$$

Whakarūnātia.

$$x-\frac{15}{2}=\frac{\sqrt{177}}{2} x-\frac{15}{2}=-\frac{\sqrt{177}}{2}$$

Me tāpiri \frac{15}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{177}+15}{2} x=\frac{15-\sqrt{177}}{2}$$