Whakaoti i te (x-60)(120-0.5x)=1600

$$( x - 60 ) ( 120 - 0.5 x ) = 1600$$

Whakaoti mō x

$$x=80$$

$$x=220$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-60 ki te 120-0.5x ka whakakotahi i ngā kupu rite.

$$150x-0.5x^{2}-7200=1600$$

Tangohia te 1600 mai i ngā taha e rua.

$$150x-0.5x^{2}-7200-1600=0$$

Tangohia te 1600 i te -7200, ka -8800.

$$150x-0.5x^{2}-8800=0$$

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.

$$-0.5x^{2}+150x-8800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -0.5 mō a, 150 mō b, me -8800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\left(-0.5\right)\left(-8800\right)}}{2\left(-0.5\right)}$$

Pūrua 150.

$$x=\frac{-150±\sqrt{22500-4\left(-0.5\right)\left(-8800\right)}}{2\left(-0.5\right)}$$

Whakareatia -4 ki te -0.5.

$$x=\frac{-150±\sqrt{22500+2\left(-8800\right)}}{2\left(-0.5\right)}$$

Whakareatia 2 ki te -8800.

$$x=\frac{-150±\sqrt{22500-17600}}{2\left(-0.5\right)}$$

Tāpiri 22500 ki te -17600.

$$x=\frac{-150±\sqrt{4900}}{2\left(-0.5\right)}$$

Tuhia te pūtakerua o te 4900.

$$x=\frac{-150±70}{2\left(-0.5\right)}$$

Whakareatia 2 ki te -0.5.

$$x=\frac{-150±70}{-1}$$

Nā, me whakaoti te whārite x=\frac{-150±70}{-1} ina he tāpiri te ±. Tāpiri -150 ki te 70.

$$x=-\frac{80}{-1}$$

Whakawehe -80 ki te -1.

$$x=80$$

Nā, me whakaoti te whārite x=\frac{-150±70}{-1} ina he tango te ±. Tango 70 mai i -150.

$$x=-\frac{220}{-1}$$

Whakawehe -220 ki te -1.

$$x=220$$

Kua oti te whārite te whakatau.

$$x=80 x=220$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te x-60 ki te 120-0.5x ka whakakotahi i ngā kupu rite.

$$150x-0.5x^{2}-7200=1600$$

Me tāpiri te 7200 ki ngā taha e rua.

$$150x-0.5x^{2}=1600+7200$$

Tāpirihia te 1600 ki te 7200, ka 8800.

$$150x-0.5x^{2}=8800$$

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.

$$-0.5x^{2}+150x=8800$$

Me whakarea ngā taha e rua ki te -2.

$$\frac{-0.5x^{2}+150x}{-0.5}=\frac{8800}{-0.5}$$

Mā te whakawehe ki te -0.5 ka wetekia te whakareanga ki te -0.5.

$$x^{2}+\frac{150}{-0.5}x=\frac{8800}{-0.5}$$

Whakawehe 150 ki te -0.5 mā te whakarea 150 ki te tau huripoki o -0.5.

$$x^{2}-300x=\frac{8800}{-0.5}$$

Whakawehe 8800 ki te -0.5 mā te whakarea 8800 ki te tau huripoki o -0.5.

$$x^{2}-300x=-17600$$

Whakawehea te -300, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -150. Nā, tāpiria te pūrua o te -150 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-300x+\left(-150\right)^{2}=-17600+\left(-150\right)^{2}$$

Pūrua -150.

$$x^{2}-300x+22500=-17600+22500$$

Tāpiri -17600 ki te 22500.

$$x^{2}-300x+22500=4900$$

Tauwehea x^{2}-300x+22500. 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}.

$$\left(x-150\right)^{2}=4900$$

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

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{4900}$$

Whakarūnātia.

$$x-150=70 x-150=-70$$

Me tāpiri 150 ki ngā taha e rua o te whārite.

$$x=220 x=80$$