Whakaoti i te 3000=(125+x)(45-x)

$$3000=(125+x)(45-x)$$

Whakaoti mō x

$$x=-105$$

$$x=25$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 125+x ki te 45-x ka whakakotahi i ngā kupu rite.

$$3000=5625-80x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$5625-80x-x^{2}=3000$$

Tangohia te 3000 mai i ngā taha e rua.

$$5625-80x-x^{2}-3000=0$$

Tangohia te 3000 i te 5625, ka 2625.

$$2625-80x-x^{2}=0$$

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.

$$-x^{2}-80x+2625=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -80 mō b, me 2625 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\left(-1\right)\times 2625}}{2\left(-1\right)}$$

Pūrua -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\left(-1\right)\times 2625}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-80\right)±\sqrt{6400+4\times 2625}}{2\left(-1\right)}$$

Whakareatia 4 ki te 2625.

$$x=\frac{-\left(-80\right)±\sqrt{6400+10500}}{2\left(-1\right)}$$

Tāpiri 6400 ki te 10500.

$$x=\frac{-\left(-80\right)±\sqrt{16900}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 16900.

$$x=\frac{-\left(-80\right)±130}{2\left(-1\right)}$$

Ko te tauaro o -80 ko 80.

$$x=\frac{80±130}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{80±130}{-2}$$

Nā, me whakaoti te whārite x=\frac{80±130}{-2} ina he tāpiri te ±. Tāpiri 80 ki te 130.

$$x=\frac{210}{-2}$$

Whakawehe 210 ki te -2.

$$x=-105$$

Nā, me whakaoti te whārite x=\frac{80±130}{-2} ina he tango te ±. Tango 130 mai i 80.

$$x=-\frac{50}{-2}$$

Whakawehe -50 ki te -2.

$$x=25$$

Kua oti te whārite te whakatau.

$$x=-105 x=25$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 125+x ki te 45-x ka whakakotahi i ngā kupu rite.

$$3000=5625-80x-x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$5625-80x-x^{2}=3000$$

Tangohia te 5625 mai i ngā taha e rua.

$$-80x-x^{2}=3000-5625$$

Tangohia te 5625 i te 3000, ka -2625.

$$-80x-x^{2}=-2625$$

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.

$$-x^{2}-80x=-2625$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-80x}{-1}=-\frac{2625}{-1}$$

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

$$x^{2}+\left(-\frac{80}{-1}\right)x=-\frac{2625}{-1}$$

Whakawehe -80 ki te -1.

$$x^{2}+80x=-\frac{2625}{-1}$$

Whakawehe -2625 ki te -1.

$$x^{2}+80x=2625$$

Whakawehea te 80, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 40. Nā, tāpiria te pūrua o te 40 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+80x+40^{2}=2625+40^{2}$$

Pūrua 40.

$$x^{2}+80x+1600=2625+1600$$

Tāpiri 2625 ki te 1600.

$$x^{2}+80x+1600=4225$$

Tauwehea x^{2}+80x+1600. 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}.

$$\left(x+40\right)^{2}=4225$$

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

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{4225}$$

Whakarūnātia.

$$x+40=65 x+40=-65$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$x=25 x=-105$$