Whakaoti i te (20-3x)(12-2x)=112

$$(20-3x)(12-2x)=112$$

Whakaoti mō x

$$x=2$$

$$x = \frac{32}{3} = 10\frac{2}{3} \approx 10.666666667$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-3x ki te 12-2x ka whakakotahi i ngā kupu rite.

$$240-76x+6x^{2}=112$$

Tangohia te 112 mai i ngā taha e rua.

$$240-76x+6x^{2}-112=0$$

Tangohia te 112 i te 240, ka 128.

$$128-76x+6x^{2}=0$$

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.

$$6x^{2}-76x+128=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -76 mō b, me 128 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-76\right)±\sqrt{\left(-76\right)^{2}-4\times 6\times 128}}{2\times 6}$$

Pūrua -76.

$$x=\frac{-\left(-76\right)±\sqrt{5776-4\times 6\times 128}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-76\right)±\sqrt{5776-24\times 128}}{2\times 6}$$

Whakareatia -24 ki te 128.

$$x=\frac{-\left(-76\right)±\sqrt{5776-3072}}{2\times 6}$$

Tāpiri 5776 ki te -3072.

$$x=\frac{-\left(-76\right)±\sqrt{2704}}{2\times 6}$$

Tuhia te pūtakerua o te 2704.

$$x=\frac{-\left(-76\right)±52}{2\times 6}$$

Ko te tauaro o -76 ko 76.

$$x=\frac{76±52}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{76±52}{12}$$

Nā, me whakaoti te whārite x=\frac{76±52}{12} ina he tāpiri te ±. Tāpiri 76 ki te 52.

$$x=\frac{128}{12}$$

Whakahekea te hautanga \frac{128}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{32}{3}$$

Nā, me whakaoti te whārite x=\frac{76±52}{12} ina he tango te ±. Tango 52 mai i 76.

$$x=\frac{24}{12}$$

Whakawehe 24 ki te 12.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=\frac{32}{3} x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 20-3x ki te 12-2x ka whakakotahi i ngā kupu rite.

$$240-76x+6x^{2}=112$$

Tangohia te 240 mai i ngā taha e rua.

$$-76x+6x^{2}=112-240$$

Tangohia te 240 i te 112, ka -128.

$$-76x+6x^{2}=-128$$

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.

$$6x^{2}-76x=-128$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-76x}{6}=-\frac{128}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\left(-\frac{76}{6}\right)x=-\frac{128}{6}$$

Whakahekea te hautanga \frac{-76}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{38}{3}x=-\frac{128}{6}$$

Whakahekea te hautanga \frac{-128}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{38}{3}x=-\frac{64}{3}$$

Whakawehea te -\frac{38}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{19}{3}. Nā, tāpiria te pūrua o te -\frac{19}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{38}{3}x+\left(-\frac{19}{3}\right)^{2}=-\frac{64}{3}+\left(-\frac{19}{3}\right)^{2}$$

Pūruatia -\frac{19}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=-\frac{64}{3}+\frac{361}{9}$$

Tāpiri -\frac{64}{3} ki te \frac{361}{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.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=\frac{169}{9}$$

Tauwehea x^{2}-\frac{38}{3}x+\frac{361}{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}.

$$\left(x-\frac{19}{3}\right)^{2}=\frac{169}{9}$$

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

$$\sqrt{\left(x-\frac{19}{3}\right)^{2}}=\sqrt{\frac{169}{9}}$$

Whakarūnātia.

$$x-\frac{19}{3}=\frac{13}{3} x-\frac{19}{3}=-\frac{13}{3}$$

Me tāpiri \frac{19}{3} ki ngā taha e rua o te whārite.

$$x=\frac{32}{3} x=2$$