Whakaoti i te 3x(11-2x)=15

$$3x(11-2x)=15$$

Whakaoti mō x

$$x=5$$

$$x=\frac{1}{2}=0.5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te 11-2x.

$$33x-6x^{2}=15$$

Tangohia te 15 mai i ngā taha e rua.

$$33x-6x^{2}-15=0$$

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.

$$-6x^{2}+33x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 33 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-6\right)\left(-15\right)}}{2\left(-6\right)}$$

Pūrua 33.

$$x=\frac{-33±\sqrt{1089-4\left(-6\right)\left(-15\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-33±\sqrt{1089+24\left(-15\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -15.

$$x=\frac{-33±\sqrt{1089-360}}{2\left(-6\right)}$$

Tāpiri 1089 ki te -360.

$$x=\frac{-33±\sqrt{729}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-33±27}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-33±27}{-12}$$

Nā, me whakaoti te whārite x=\frac{-33±27}{-12} ina he tāpiri te ±. Tāpiri -33 ki te 27.

$$x=-\frac{6}{-12}$$

Whakahekea te hautanga \frac{-6}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-33±27}{-12} ina he tango te ±. Tango 27 mai i -33.

$$x=-\frac{60}{-12}$$

Whakawehe -60 ki te -12.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{2} x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 3x ki te 11-2x.

$$33x-6x^{2}=15$$

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.

$$-6x^{2}+33x=15$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+33x}{-6}=\frac{15}{-6}$$

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

$$x^{2}+\frac{33}{-6}x=\frac{15}{-6}$$

Whakahekea te hautanga \frac{33}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{11}{2}x=\frac{15}{-6}$$

Whakahekea te hautanga \frac{15}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{11}{2}x=-\frac{5}{2}$$

Whakawehea te -\frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{4}. Nā, tāpiria te pūrua o te -\frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{5}{2}+\left(-\frac{11}{4}\right)^{2}$$

Pūruatia -\frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{5}{2}+\frac{121}{16}$$

Tāpiri -\frac{5}{2} ki te \frac{121}{16} 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.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{81}{16}$$

Tauwehea x^{2}-\frac{11}{2}x+\frac{121}{16}. 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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$x-\frac{11}{4}=\frac{9}{4} x-\frac{11}{4}=-\frac{9}{4}$$

Me tāpiri \frac{11}{4} ki ngā taha e rua o te whārite.

$$x=5 x=\frac{1}{2}$$