Whakaoti i te 42=(2x)(x+9)

$$42=(2x)(x+9)$$

Whakaoti mō x

$$x = \frac{\sqrt{165} - 9}{2} \approx 1.922616289$$

$$x=\frac{-\sqrt{165}-9}{2}\approx -10.922616289$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+9.

$$42=2x^{2}+18x$$

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

$$2x^{2}+18x=42$$

Tangohia te 42 mai i ngā taha e rua.

$$2x^{2}+18x-42=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 18 mō b, me -42 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 2\left(-42\right)}}{2\times 2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 2\left(-42\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-18±\sqrt{324-8\left(-42\right)}}{2\times 2}$$

Whakareatia -8 ki te -42.

$$x=\frac{-18±\sqrt{324+336}}{2\times 2}$$

Tāpiri 324 ki te 336.

$$x=\frac{-18±\sqrt{660}}{2\times 2}$$

Tuhia te pūtakerua o te 660.

$$x=\frac{-18±2\sqrt{165}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-18±2\sqrt{165}}{4}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{165}}{4} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{165}.

$$x=\frac{2\sqrt{165}-18}{4}$$

Whakawehe -18+2\sqrt{165} ki te 4.

$$x=\frac{\sqrt{165}-9}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{165}}{4} ina he tango te ±. Tango 2\sqrt{165} mai i -18.

$$x=\frac{-2\sqrt{165}-18}{4}$$

Whakawehe -18-2\sqrt{165} ki te 4.

$$x=\frac{-\sqrt{165}-9}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{165}-9}{2} x=\frac{-\sqrt{165}-9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+9.

$$42=2x^{2}+18x$$

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

$$2x^{2}+18x=42$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+18x}{2}=\frac{42}{2}$$

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

$$x^{2}+\frac{18}{2}x=\frac{42}{2}$$

Whakawehe 18 ki te 2.

$$x^{2}+9x=\frac{42}{2}$$

Whakawehe 42 ki te 2.

$$x^{2}+9x=21$$

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

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=21+\left(\frac{9}{2}\right)^{2}$$

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+9x+\frac{81}{4}=21+\frac{81}{4}$$

Tāpiri 21 ki te \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{165}{4}$$

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

$$\left(x+\frac{9}{2}\right)^{2}=\frac{165}{4}$$

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

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{165}{4}}$$

Whakarūnātia.

$$x+\frac{9}{2}=\frac{\sqrt{165}}{2} x+\frac{9}{2}=-\frac{\sqrt{165}}{2}$$

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{165}-9}{2} x=\frac{-\sqrt{165}-9}{2}$$