Whakaoti i te (20+x)(100-x)=2375

$$(20+x)(100-x)=2375$$

Whakaoti mō x

$$x=5$$

$$x=75$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$2000+80x-x^{2}=2375$$

Tangohia te 2375 mai i ngā taha e rua.

$$2000+80x-x^{2}-2375=0$$

Tangohia te 2375 i te 2000, ka -375.

$$-375+80x-x^{2}=0$$

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.

$$-x^{2}+80x-375=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 80 mō b, me -375 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\left(-1\right)\left(-375\right)}}{2\left(-1\right)}$$

Pūrua 80.

$$x=\frac{-80±\sqrt{6400-4\left(-1\right)\left(-375\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-80±\sqrt{6400+4\left(-375\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -375.

$$x=\frac{-80±\sqrt{6400-1500}}{2\left(-1\right)}$$

Tāpiri 6400 ki te -1500.

$$x=\frac{-80±\sqrt{4900}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 4900.

$$x=\frac{-80±70}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-80±70}{-2}$$

Nā, me whakaoti te whārite x=\frac{-80±70}{-2} ina he tāpiri te ±. Tāpiri -80 ki te 70.

$$x=-\frac{10}{-2}$$

Whakawehe -10 ki te -2.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-80±70}{-2} ina he tango te ±. Tango 70 mai i -80.

$$x=-\frac{150}{-2}$$

Whakawehe -150 ki te -2.

$$x=75$$

Kua oti te whārite te whakatau.

$$x=5 x=75$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$2000+80x-x^{2}=2375$$

Tangohia te 2000 mai i ngā taha e rua.

$$80x-x^{2}=2375-2000$$

Tangohia te 2000 i te 2375, ka 375.

$$80x-x^{2}=375$$

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.

$$-x^{2}+80x=375$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+80x}{-1}=\frac{375}{-1}$$

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

$$x^{2}+\frac{80}{-1}x=\frac{375}{-1}$$

Whakawehe 80 ki te -1.

$$x^{2}-80x=\frac{375}{-1}$$

Whakawehe 375 ki te -1.

$$x^{2}-80x=-375$$

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.

$$x^{2}-80x+\left(-40\right)^{2}=-375+\left(-40\right)^{2}$$

Pūrua -40.

$$x^{2}-80x+1600=-375+1600$$

Tāpiri -375 ki te 1600.

$$x^{2}-80x+1600=1225$$

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}.

$$\left(x-40\right)^{2}=1225$$

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

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{1225}$$

Whakarūnātia.

$$x-40=35 x-40=-35$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$x=75 x=5$$