Whakaoti i te 450pi=2pix(x+15)

$$450 \pi =2 \pi x(x+15)$$

Whakaoti mō x

$$x = \frac{15 \sqrt{5} - 15}{2} \approx 9.270509831$$

$$x=\frac{-15\sqrt{5}-15}{2}\approx -24.270509831$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakakore te \pi ki ngā taha e rua.

$$450=2x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+15.

$$450=2x^{2}+30x$$

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

$$2x^{2}+30x=450$$

Tangohia te 450 mai i ngā taha e rua.

$$2x^{2}+30x-450=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 30 mō b, me -450 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2\left(-450\right)}}{2\times 2}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\times 2\left(-450\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-30±\sqrt{900-8\left(-450\right)}}{2\times 2}$$

Whakareatia -8 ki te -450.

$$x=\frac{-30±\sqrt{900+3600}}{2\times 2}$$

Tāpiri 900 ki te 3600.

$$x=\frac{-30±\sqrt{4500}}{2\times 2}$$

Tuhia te pūtakerua o te 4500.

$$x=\frac{-30±30\sqrt{5}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-30±30\sqrt{5}}{4}$$

Nā, me whakaoti te whārite x=\frac{-30±30\sqrt{5}}{4} ina he tāpiri te ±. Tāpiri -30 ki te 30\sqrt{5}.

$$x=\frac{30\sqrt{5}-30}{4}$$

Whakawehe -30+30\sqrt{5} ki te 4.

$$x=\frac{15\sqrt{5}-15}{2}$$

Nā, me whakaoti te whārite x=\frac{-30±30\sqrt{5}}{4} ina he tango te ±. Tango 30\sqrt{5} mai i -30.

$$x=\frac{-30\sqrt{5}-30}{4}$$

Whakawehe -30-30\sqrt{5} ki te 4.

$$x=\frac{-15\sqrt{5}-15}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakakore te \pi ki ngā taha e rua.

$$450=2x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+15.

$$450=2x^{2}+30x$$

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

$$2x^{2}+30x=450$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+30x}{2}=\frac{450}{2}$$

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

$$x^{2}+\frac{30}{2}x=\frac{450}{2}$$

Whakawehe 30 ki te 2.

$$x^{2}+15x=\frac{450}{2}$$

Whakawehe 450 ki te 2.

$$x^{2}+15x=225$$

Whakawehea te 15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{2}. Nā, tāpiria te pūrua o te \frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=225+\left(\frac{15}{2}\right)^{2}$$

Pūruatia \frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+15x+\frac{225}{4}=225+\frac{225}{4}$$

Tāpiri 225 ki te \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{1125}{4}$$

Tauwehea x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{1125}{4}$$

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

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1125}{4}}$$

Whakarūnātia.

$$x+\frac{15}{2}=\frac{15\sqrt{5}}{2} x+\frac{15}{2}=-\frac{15\sqrt{5}}{2}$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$