Whakaoti i te (3x+5)(x-2)=14

$$(3x+5)(x-2)=14$$

Whakaoti mō x

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

$$x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x+5 ki te x-2 ka whakakotahi i ngā kupu rite.

$$3x^{2}-x-10=14$$

Tangohia te 14 mai i ngā taha e rua.

$$3x^{2}-x-10-14=0$$

Tangohia te 14 i te -10, ka -24.

$$3x^{2}-x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -1 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-24\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-1\right)±\sqrt{1-12\left(-24\right)}}{2\times 3}$$

Whakareatia -12 ki te -24.

$$x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 3}$$

Tāpiri 1 ki te 288.

$$x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 3}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-\left(-1\right)±17}{2\times 3}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±17}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{1±17}{6}$$

Nā, me whakaoti te whārite x=\frac{1±17}{6} ina he tāpiri te ±. Tāpiri 1 ki te 17.

$$x=\frac{18}{6}$$

Whakawehe 18 ki te 6.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{1±17}{6} ina he tango te ±. Tango 17 mai i 1.

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

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.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x+5 ki te x-2 ka whakakotahi i ngā kupu rite.

$$3x^{2}-x-10=14$$

Me tāpiri te 10 ki ngā taha e rua.

$$3x^{2}-x=14+10$$

Tāpirihia te 14 ki te 10, ka 24.

$$3x^{2}-x=24$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-x}{3}=\frac{24}{3}$$

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

$$x^{2}-\frac{1}{3}x=\frac{24}{3}$$

Whakawehe 24 ki te 3.

$$x^{2}-\frac{1}{3}x=8$$

Whakawehea te -\frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{6}. Nā, tāpiria te pūrua o te -\frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=8+\left(-\frac{1}{6}\right)^{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=8+\frac{1}{36}$$

Tāpiri 8 ki te \frac{1}{36}.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{289}{36}$$

Tauwehea x^{2}-\frac{1}{3}x+\frac{1}{36}. 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}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{289}{36}$$

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

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{289}{36}}$$

Whakarūnātia.

$$x-\frac{1}{6}=\frac{17}{6} x-\frac{1}{6}=-\frac{17}{6}$$

Me tāpiri \frac{1}{6} ki ngā taha e rua o te whārite.

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