Whakaoti i te 3:7=(x+1):2x

$$3 : 7 = ( x + 1 ) : 2 x$$

Whakaoti mō x

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}\approx 0.552208562$$

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}\approx -1.552208562$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 14, arā, te tauraro pātahi he tino iti rawa te kitea o 7,2.

$$6=7\left(x+1\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x+1.

$$6=\left(7x+7\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7x+7 ki te x.

$$6=7x^{2}+7x$$

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

$$7x^{2}+7x=6$$

Tangohia te 6 mai i ngā taha e rua.

$$7x^{2}+7x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, 7 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 7\left(-6\right)}}{2\times 7}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 7\left(-6\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-7±\sqrt{49-28\left(-6\right)}}{2\times 7}$$

Whakareatia -28 ki te -6.

$$x=\frac{-7±\sqrt{49+168}}{2\times 7}$$

Tāpiri 49 ki te 168.

$$x=\frac{-7±\sqrt{217}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-7±\sqrt{217}}{14}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{217}}{14} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{217}.

$$x=\frac{\sqrt{217}-7}{14}$$

Whakawehe -7+\sqrt{217} ki te 14.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{217}}{14} ina he tango te ±. Tango \sqrt{217} mai i -7.

$$x=\frac{-\sqrt{217}-7}{14}$$

Whakawehe -7-\sqrt{217} ki te 14.

$$x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2} x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 14, arā, te tauraro pātahi he tino iti rawa te kitea o 7,2.

$$6=7\left(x+1\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x+1.

$$6=\left(7x+7\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7x+7 ki te x.

$$6=7x^{2}+7x$$

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

$$7x^{2}+7x=6$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}+7x}{7}=\frac{6}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}+\frac{7}{7}x=\frac{6}{7}$$

Whakawehe 7 ki te 7.

$$x^{2}+x=\frac{6}{7}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{6}{7}+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=\frac{6}{7}+\frac{1}{4}$$

Tāpiri \frac{6}{7} ki te \frac{1}{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.

$$x^{2}+x+\frac{1}{4}=\frac{31}{28}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=\frac{31}{28}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{31}{28}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{217}}{14} x+\frac{1}{2}=-\frac{\sqrt{217}}{14}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{217}}{14}-\frac{1}{2} x=-\frac{\sqrt{217}}{14}-\frac{1}{2}$$