Whakaoti i te (2x+1)(x-2)=7

$$( 2 x + 1 ) ( x - 2 ) = 7$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+1 ki te x-2 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-2=7$$

Tangohia te 7 mai i ngā taha e rua.

$$2x^{2}-3x-2-7=0$$

Tangohia te 7 i te -2, ka -9.

$$2x^{2}-3x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -3 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-9\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-9\right)}}{2\times 2}$$

Whakareatia -8 ki te -9.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\times 2}$$

Tāpiri 9 ki te 72.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2\times 2}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-3\right)±9}{2\times 2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±9}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{3±9}{4}$$

Nā, me whakaoti te whārite x=\frac{3±9}{4} ina he tāpiri te ±. Tāpiri 3 ki te 9.

$$x=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{3±9}{4} ina he tango te ±. Tango 9 mai i 3.

$$x=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+1 ki te x-2 ka whakakotahi i ngā kupu rite.

$$2x^{2}-3x-2=7$$

Me tāpiri te 2 ki ngā taha e rua.

$$2x^{2}-3x=7+2$$

Tāpirihia te 7 ki te 2, ka 9.

$$2x^{2}-3x=9$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-3x}{2}=\frac{9}{2}$$

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

$$x^{2}-\frac{3}{2}x=\frac{9}{2}$$

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{9}{2}+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{9}{2}+\frac{9}{16}$$

Tāpiri \frac{9}{2} ki te \frac{9}{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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{81}{16}$$

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

$$\left(x-\frac{3}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{9}{4} x-\frac{3}{4}=-\frac{9}{4}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

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