Whakaoti i te (20-3x)(12-x)=112

$$( 20 - 3 x ) ( 12 - x ) = 112$$

Whakaoti mō x

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$x=16$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-3x ki te 12-x ka whakakotahi i ngā kupu rite.

$$240-56x+3x^{2}=112$$

Tangohia te 112 mai i ngā taha e rua.

$$240-56x+3x^{2}-112=0$$

Tangohia te 112 i te 240, ka 128.

$$128-56x+3x^{2}=0$$

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.

$$3x^{2}-56x+128=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -56 mō b, me 128 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 3\times 128}}{2\times 3}$$

Pūrua -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 3\times 128}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-56\right)±\sqrt{3136-12\times 128}}{2\times 3}$$

Whakareatia -12 ki te 128.

$$x=\frac{-\left(-56\right)±\sqrt{3136-1536}}{2\times 3}$$

Tāpiri 3136 ki te -1536.

$$x=\frac{-\left(-56\right)±\sqrt{1600}}{2\times 3}$$

Tuhia te pūtakerua o te 1600.

$$x=\frac{-\left(-56\right)±40}{2\times 3}$$

Ko te tauaro o -56 ko 56.

$$x=\frac{56±40}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{56±40}{6}$$

Nā, me whakaoti te whārite x=\frac{56±40}{6} ina he tāpiri te ±. Tāpiri 56 ki te 40.

$$x=\frac{96}{6}$$

Whakawehe 96 ki te 6.

$$x=16$$

Nā, me whakaoti te whārite x=\frac{56±40}{6} ina he tango te ±. Tango 40 mai i 56.

$$x=\frac{16}{6}$$

Whakahekea te hautanga \frac{16}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{8}{3}$$

Kua oti te whārite te whakatau.

$$x=16 x=\frac{8}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-3x ki te 12-x ka whakakotahi i ngā kupu rite.

$$240-56x+3x^{2}=112$$

Tangohia te 240 mai i ngā taha e rua.

$$-56x+3x^{2}=112-240$$

Tangohia te 240 i te 112, ka -128.

$$-56x+3x^{2}=-128$$

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.

$$3x^{2}-56x=-128$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-56x}{3}=-\frac{128}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{56}{3}x=-\frac{128}{3}$$

Whakawehea te -\frac{56}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{28}{3}. Nā, tāpiria te pūrua o te -\frac{28}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{56}{3}x+\left(-\frac{28}{3}\right)^{2}=-\frac{128}{3}+\left(-\frac{28}{3}\right)^{2}$$

Pūruatia -\frac{28}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{56}{3}x+\frac{784}{9}=-\frac{128}{3}+\frac{784}{9}$$

Tāpiri -\frac{128}{3} ki te \frac{784}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{56}{3}x+\frac{784}{9}=\frac{400}{9}$$

Tauwehea x^{2}-\frac{56}{3}x+\frac{784}{9}. 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}.

$$\left(x-\frac{28}{3}\right)^{2}=\frac{400}{9}$$

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

$$\sqrt{\left(x-\frac{28}{3}\right)^{2}}=\sqrt{\frac{400}{9}}$$

Whakarūnātia.

$$x-\frac{28}{3}=\frac{20}{3} x-\frac{28}{3}=-\frac{20}{3}$$

Me tāpiri \frac{28}{3} ki ngā taha e rua o te whārite.

$$x=16 x=\frac{8}{3}$$