Whakaoti i te (2x+3)(x-1)=15

$$(2x+3)(x-1)=15$$

Whakaoti mō x

$$x = \frac{\sqrt{145} - 1}{4} \approx 2.760398645$$

$$x=\frac{-\sqrt{145}-1}{4}\approx -3.260398645$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$2x^{2}+x-3=15$$

Tangohia te 15 mai i ngā taha e rua.

$$2x^{2}+x-3-15=0$$

Tangohia te 15 i te -3, ka -18.

$$2x^{2}+x-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 1 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-18\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-1±\sqrt{1-8\left(-18\right)}}{2\times 2}$$

Whakareatia -8 ki te -18.

$$x=\frac{-1±\sqrt{1+144}}{2\times 2}$$

Tāpiri 1 ki te 144.

$$x=\frac{-1±\sqrt{145}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-1±\sqrt{145}}{4}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{145}}{4} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{145}.

$$x=\frac{\sqrt{145}-1}{4}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{145}}{4} ina he tango te ±. Tango \sqrt{145} mai i -1.

$$x=\frac{-\sqrt{145}-1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{145}-1}{4} x=\frac{-\sqrt{145}-1}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$2x^{2}+x-3=15$$

Me tāpiri te 3 ki ngā taha e rua.

$$2x^{2}+x=15+3$$

Tāpirihia te 15 ki te 3, ka 18.

$$2x^{2}+x=18$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+x}{2}=\frac{18}{2}$$

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

$$x^{2}+\frac{1}{2}x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x^{2}+\frac{1}{2}x=9$$

Whakawehea te \frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{4}. Nā, tāpiria te pūrua o te \frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=9+\left(\frac{1}{4}\right)^{2}$$

Pūruatia \frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=9+\frac{1}{16}$$

Tāpiri 9 ki te \frac{1}{16}.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{145}{16}$$

Tauwehea x^{2}+\frac{1}{2}x+\frac{1}{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}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{145}{16}$$

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

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{145}{16}}$$

Whakarūnātia.

$$x+\frac{1}{4}=\frac{\sqrt{145}}{4} x+\frac{1}{4}=-\frac{\sqrt{145}}{4}$$

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{145}-1}{4} x=\frac{-\sqrt{145}-1}{4}$$