Whakaoti i te (2x+1)(x-3)=114

$$( 2 x + 1 ) ( x - 3 ) = 114$$

Whakaoti mō x

$$x = -\frac{13}{2} = -6\frac{1}{2} = -6.5$$

$$x=9$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+1 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x-3=114$$

Tangohia te 114 mai i ngā taha e rua.

$$2x^{2}-5x-3-114=0$$

Tangohia te 114 i te -3, ka -117.

$$2x^{2}-5x-117=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -5 mō b, me -117 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-117\right)}}{2\times 2}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-117\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-117\right)}}{2\times 2}$$

Whakareatia -8 ki te -117.

$$x=\frac{-\left(-5\right)±\sqrt{25+936}}{2\times 2}$$

Tāpiri 25 ki te 936.

$$x=\frac{-\left(-5\right)±\sqrt{961}}{2\times 2}$$

Tuhia te pūtakerua o te 961.

$$x=\frac{-\left(-5\right)±31}{2\times 2}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±31}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{5±31}{4}$$

Nā, me whakaoti te whārite x=\frac{5±31}{4} ina he tāpiri te ±. Tāpiri 5 ki te 31.

$$x=\frac{36}{4}$$

Whakawehe 36 ki te 4.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{5±31}{4} ina he tango te ±. Tango 31 mai i 5.

$$x=-\frac{26}{4}$$

Whakahekea te hautanga \frac{-26}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{13}{2}$$

Kua oti te whārite te whakatau.

$$x=9 x=-\frac{13}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+1 ki te x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x-3=114$$

Me tāpiri te 3 ki ngā taha e rua.

$$2x^{2}-5x=114+3$$

Tāpirihia te 114 ki te 3, ka 117.

$$2x^{2}-5x=117$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-5x}{2}=\frac{117}{2}$$

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

$$x^{2}-\frac{5}{2}x=\frac{117}{2}$$

Whakawehea te -\frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{4}. Nā, tāpiria te pūrua o te -\frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=\frac{117}{2}+\left(-\frac{5}{4}\right)^{2}$$

Pūruatia -\frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{117}{2}+\frac{25}{16}$$

Tāpiri \frac{117}{2} ki te \frac{25}{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.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{961}{16}$$

Tauwehea x^{2}-\frac{5}{2}x+\frac{25}{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}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{961}{16}$$

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

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{961}{16}}$$

Whakarūnātia.

$$x-\frac{5}{4}=\frac{31}{4} x-\frac{5}{4}=-\frac{31}{4}$$

Me tāpiri \frac{5}{4} ki ngā taha e rua o te whārite.

$$x=9 x=-\frac{13}{2}$$