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

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

Whakaoti mō x

$$x=\frac{\sqrt{19}-3}{2}\approx 0.679449472$$

$$x=\frac{-\sqrt{19}-3}{2}\approx -3.679449472$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+3.

$$2x^{2}+6x=5$$

Tangohia te 5 mai i ngā taha e rua.

$$2x^{2}+6x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 6 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 2\left(-5\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-6±\sqrt{36-8\left(-5\right)}}{2\times 2}$$

Whakareatia -8 ki te -5.

$$x=\frac{-6±\sqrt{36+40}}{2\times 2}$$

Tāpiri 36 ki te 40.

$$x=\frac{-6±\sqrt{76}}{2\times 2}$$

Tuhia te pūtakerua o te 76.

$$x=\frac{-6±2\sqrt{19}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-6±2\sqrt{19}}{4}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{19}}{4} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-6}{4}$$

Whakawehe -6+2\sqrt{19} ki te 4.

$$x=\frac{\sqrt{19}-3}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{19}}{4} ina he tango te ±. Tango 2\sqrt{19} mai i -6.

$$x=\frac{-2\sqrt{19}-6}{4}$$

Whakawehe -6-2\sqrt{19} ki te 4.

$$x=\frac{-\sqrt{19}-3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{19}-3}{2} x=\frac{-\sqrt{19}-3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+3.

$$2x^{2}+6x=5$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+6x}{2}=\frac{5}{2}$$

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

$$x^{2}+\frac{6}{2}x=\frac{5}{2}$$

Whakawehe 6 ki te 2.

$$x^{2}+3x=\frac{5}{2}$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{5}{2}+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=\frac{5}{2}+\frac{9}{4}$$

Tāpiri \frac{5}{2} ki te \frac{9}{4} 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.

$$x^{2}+3x+\frac{9}{4}=\frac{19}{4}$$

Tauwehea x^{2}+3x+\frac{9}{4}. 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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{19}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{\sqrt{19}}{2} x+\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{19}-3}{2} x=\frac{-\sqrt{19}-3}{2}$$