Whakaoti i te 3w^2+15w+12=w

$$3 w ^ { 2 } + 15 w + 12 = w$$

Whakaoti mō w

$$w=\frac{\sqrt{13}-7}{3}\approx -1.131482908$$

$$w=\frac{-\sqrt{13}-7}{3}\approx -3.535183758$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te w mai i ngā taha e rua.

$$3w^{2}+15w+12-w=0$$

Pahekotia te 15w me -w, ka 14w.

$$3w^{2}+14w+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 14 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-14±\sqrt{14^{2}-4\times 3\times 12}}{2\times 3}$$

Pūrua 14.

$$w=\frac{-14±\sqrt{196-4\times 3\times 12}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$w=\frac{-14±\sqrt{196-12\times 12}}{2\times 3}$$

Whakareatia -12 ki te 12.

$$w=\frac{-14±\sqrt{196-144}}{2\times 3}$$

Tāpiri 196 ki te -144.

$$w=\frac{-14±\sqrt{52}}{2\times 3}$$

Tuhia te pūtakerua o te 52.

$$w=\frac{-14±2\sqrt{13}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$w=\frac{-14±2\sqrt{13}}{6}$$

Nā, me whakaoti te whārite w=\frac{-14±2\sqrt{13}}{6} ina he tāpiri te ±. Tāpiri -14 ki te 2\sqrt{13}.

$$w=\frac{2\sqrt{13}-14}{6}$$

Whakawehe -14+2\sqrt{13} ki te 6.

$$w=\frac{\sqrt{13}-7}{3}$$

Nā, me whakaoti te whārite w=\frac{-14±2\sqrt{13}}{6} ina he tango te ±. Tango 2\sqrt{13} mai i -14.

$$w=\frac{-2\sqrt{13}-14}{6}$$

Whakawehe -14-2\sqrt{13} ki te 6.

$$w=\frac{-\sqrt{13}-7}{3}$$

Kua oti te whārite te whakatau.

$$w=\frac{\sqrt{13}-7}{3} w=\frac{-\sqrt{13}-7}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te w mai i ngā taha e rua.

$$3w^{2}+15w+12-w=0$$

Pahekotia te 15w me -w, ka 14w.

$$3w^{2}+14w+12=0$$

Tangohia te 12 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3w^{2}+14w=-12$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3w^{2}+14w}{3}=-\frac{12}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$w^{2}+\frac{14}{3}w=-\frac{12}{3}$$

Whakawehe -12 ki te 3.

$$w^{2}+\frac{14}{3}w=-4$$

Whakawehea te \frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{3}. Nā, tāpiria te pūrua o te \frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$w^{2}+\frac{14}{3}w+\left(\frac{7}{3}\right)^{2}=-4+\left(\frac{7}{3}\right)^{2}$$

Pūruatia \frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$w^{2}+\frac{14}{3}w+\frac{49}{9}=-4+\frac{49}{9}$$

Tāpiri -4 ki te \frac{49}{9}.

$$w^{2}+\frac{14}{3}w+\frac{49}{9}=\frac{13}{9}$$

Tauwehea w^{2}+\frac{14}{3}w+\frac{49}{9}. 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}.

$$\left(w+\frac{7}{3}\right)^{2}=\frac{13}{9}$$

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

$$\sqrt{\left(w+\frac{7}{3}\right)^{2}}=\sqrt{\frac{13}{9}}$$

Whakarūnātia.

$$w+\frac{7}{3}=\frac{\sqrt{13}}{3} w+\frac{7}{3}=-\frac{\sqrt{13}}{3}$$

Me tango \frac{7}{3} mai i ngā taha e rua o te whārite.

$$w=\frac{\sqrt{13}-7}{3} w=\frac{-\sqrt{13}-7}{3}$$