Whakaoti i te 65=2*x^2+9*x+5=

$$65=2 \times { x }^{ 2 } +9 \times x+5=$$

Whakaoti mō x

$$x = \frac{\sqrt{561} - 9}{4} \approx 3.671359641$$

$$x=\frac{-\sqrt{561}-9}{4}\approx -8.171359641$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}+9x+5=65$$

Tangohia te 65 mai i ngā taha e rua.

$$2x^{2}+9x+5-65=0$$

Tangohia te 65 i te 5, ka -60.

$$2x^{2}+9x-60=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 9 mō b, me -60 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 2\left(-60\right)}}{2\times 2}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 2\left(-60\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-9±\sqrt{81-8\left(-60\right)}}{2\times 2}$$

Whakareatia -8 ki te -60.

$$x=\frac{-9±\sqrt{81+480}}{2\times 2}$$

Tāpiri 81 ki te 480.

$$x=\frac{-9±\sqrt{561}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-9±\sqrt{561}}{4}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{561}}{4} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{561}.

$$x=\frac{\sqrt{561}-9}{4}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{561}}{4} ina he tango te ±. Tango \sqrt{561} mai i -9.

$$x=\frac{-\sqrt{561}-9}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{561}-9}{4} x=\frac{-\sqrt{561}-9}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$2x^{2}+9x+5=65$$

Tangohia te 5 mai i ngā taha e rua.

$$2x^{2}+9x=65-5$$

Tangohia te 5 i te 65, ka 60.

$$2x^{2}+9x=60$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+9x}{2}=\frac{60}{2}$$

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

$$x^{2}+\frac{9}{2}x=\frac{60}{2}$$

Whakawehe 60 ki te 2.

$$x^{2}+\frac{9}{2}x=30$$

Whakawehea te \frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{4}. Nā, tāpiria te pūrua o te \frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=30+\left(\frac{9}{4}\right)^{2}$$

Pūruatia \frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=30+\frac{81}{16}$$

Tāpiri 30 ki te \frac{81}{16}.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{561}{16}$$

Tauwehea x^{2}+\frac{9}{2}x+\frac{81}{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}.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{561}{16}$$

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

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{561}{16}}$$

Whakarūnātia.

$$x+\frac{9}{4}=\frac{\sqrt{561}}{4} x+\frac{9}{4}=-\frac{\sqrt{561}}{4}$$

Me tango \frac{9}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{561}-9}{4} x=\frac{-\sqrt{561}-9}{4}$$