Whakaoti i te (200-x)(20+2x)=1750

$$(200-x)(20+2x)=1750$$

Whakaoti mō x

$$x=5\sqrt{406}+95\approx 195.747208398$$

$$x=95-5\sqrt{406}\approx -5.747208398$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 200-x ki te 20+2x ka whakakotahi i ngā kupu rite.

$$4000+380x-2x^{2}=1750$$

Tangohia te 1750 mai i ngā taha e rua.

$$4000+380x-2x^{2}-1750=0$$

Tangohia te 1750 i te 4000, ka 2250.

$$2250+380x-2x^{2}=0$$

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.

$$-2x^{2}+380x+2250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 380 mō b, me 2250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-380±\sqrt{380^{2}-4\left(-2\right)\times 2250}}{2\left(-2\right)}$$

Pūrua 380.

$$x=\frac{-380±\sqrt{144400-4\left(-2\right)\times 2250}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-380±\sqrt{144400+8\times 2250}}{2\left(-2\right)}$$

Whakareatia 8 ki te 2250.

$$x=\frac{-380±\sqrt{144400+18000}}{2\left(-2\right)}$$

Tāpiri 144400 ki te 18000.

$$x=\frac{-380±\sqrt{162400}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 162400.

$$x=\frac{-380±20\sqrt{406}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-380±20\sqrt{406}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-380±20\sqrt{406}}{-4} ina he tāpiri te ±. Tāpiri -380 ki te 20\sqrt{406}.

$$x=\frac{20\sqrt{406}-380}{-4}$$

Whakawehe -380+20\sqrt{406} ki te -4.

$$x=95-5\sqrt{406}$$

Nā, me whakaoti te whārite x=\frac{-380±20\sqrt{406}}{-4} ina he tango te ±. Tango 20\sqrt{406} mai i -380.

$$x=\frac{-20\sqrt{406}-380}{-4}$$

Whakawehe -380-20\sqrt{406} ki te -4.

$$x=5\sqrt{406}+95$$

Kua oti te whārite te whakatau.

$$x=95-5\sqrt{406} x=5\sqrt{406}+95$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 200-x ki te 20+2x ka whakakotahi i ngā kupu rite.

$$4000+380x-2x^{2}=1750$$

Tangohia te 4000 mai i ngā taha e rua.

$$380x-2x^{2}=1750-4000$$

Tangohia te 4000 i te 1750, ka -2250.

$$380x-2x^{2}=-2250$$

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.

$$-2x^{2}+380x=-2250$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+380x}{-2}=-\frac{2250}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{380}{-2}x=-\frac{2250}{-2}$$

Whakawehe 380 ki te -2.

$$x^{2}-190x=-\frac{2250}{-2}$$

Whakawehe -2250 ki te -2.

$$x^{2}-190x=1125$$

Whakawehea te -190, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -95. Nā, tāpiria te pūrua o te -95 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-190x+\left(-95\right)^{2}=1125+\left(-95\right)^{2}$$

Pūrua -95.

$$x^{2}-190x+9025=1125+9025$$

Tāpiri 1125 ki te 9025.

$$x^{2}-190x+9025=10150$$

Tauwehea x^{2}-190x+9025. 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}.

$$\left(x-95\right)^{2}=10150$$

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

$$\sqrt{\left(x-95\right)^{2}}=\sqrt{10150}$$

Whakarūnātia.

$$x-95=5\sqrt{406} x-95=-5\sqrt{406}$$

Me tāpiri 95 ki ngā taha e rua o te whārite.

$$x=5\sqrt{406}+95 x=95-5\sqrt{406}$$