Whakaoti i te (100+x)(200-x)=20000

$$( 100 + x ) ( 200 - x ) = 20000$$

Whakaoti mō x

$$x=100$$

$$x=0$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 100+x ki te 200-x ka whakakotahi i ngā kupu rite.

$$20000+100x-x^{2}=20000$$

Tangohia te 20000 mai i ngā taha e rua.

$$20000+100x-x^{2}-20000=0$$

Tangohia te 20000 i te 20000, ka 0.

$$100x-x^{2}=0$$

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.

$$-x^{2}+100x=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 100 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100^{2}.

$$x=\frac{-100±100}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-100±100}{-2}$$

Nā, me whakaoti te whārite x=\frac{-100±100}{-2} ina he tāpiri te ±. Tāpiri -100 ki te 100.

$$x=\frac{0}{-2}$$

Whakawehe 0 ki te -2.

$$x=0$$

Nā, me whakaoti te whārite x=\frac{-100±100}{-2} ina he tango te ±. Tango 100 mai i -100.

$$x=-\frac{200}{-2}$$

Whakawehe -200 ki te -2.

$$x=100$$

Kua oti te whārite te whakatau.

$$x=0 x=100$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 100+x ki te 200-x ka whakakotahi i ngā kupu rite.

$$20000+100x-x^{2}=20000$$

Tangohia te 20000 mai i ngā taha e rua.

$$100x-x^{2}=20000-20000$$

Tangohia te 20000 i te 20000, ka 0.

$$100x-x^{2}=0$$

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.

$$-x^{2}+100x=0$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+100x}{-1}=\frac{0}{-1}$$

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

$$x^{2}+\frac{100}{-1}x=\frac{0}{-1}$$

Whakawehe 100 ki te -1.

$$x^{2}-100x=\frac{0}{-1}$$

Whakawehe 0 ki te -1.

$$x^{2}-100x=0$$

Whakawehea te -100, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -50. Nā, tāpiria te pūrua o te -50 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-100x+\left(-50\right)^{2}=\left(-50\right)^{2}$$

Pūrua -50.

$$x^{2}-100x+2500=2500$$

Tauwehea te x^{2}-100x+2500. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2500$$

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

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2500}$$

Whakarūnātia.

$$x-50=50 x-50=-50$$

Me tāpiri 50 ki ngā taha e rua o te whārite.

$$x=100 x=0$$