Whakaoti i te 3x^2+7x=5

$$3 x ^ { 2 } + 7 x = 5$$

Whakaoti mō x

$$x=\frac{\sqrt{109}-7}{6}\approx 0.573384418$$

$$x=\frac{-\sqrt{109}-7}{6}\approx -2.906717751$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$3x^{2}+7x=5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$3x^{2}+7x-5=5-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$3x^{2}+7x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 7 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 3\left(-5\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-7±\sqrt{49-12\left(-5\right)}}{2\times 3}$$

Whakareatia -12 ki te -5.

$$x=\frac{-7±\sqrt{49+60}}{2\times 3}$$

Tāpiri 49 ki te 60.

$$x=\frac{-7±\sqrt{109}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-7±\sqrt{109}}{6}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{109}}{6} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{109}.

$$x=\frac{\sqrt{109}-7}{6}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{109}}{6} ina he tango te ±. Tango \sqrt{109} mai i -7.

$$x=\frac{-\sqrt{109}-7}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{109}-7}{6} x=\frac{-\sqrt{109}-7}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$3x^{2}+7x=5$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+7x}{3}=\frac{5}{3}$$

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

$$x^{2}+\frac{7}{3}x=\frac{5}{3}$$

Whakawehea te \frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{6}. Nā, tāpiria te pūrua o te \frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=\frac{5}{3}+\left(\frac{7}{6}\right)^{2}$$

Pūruatia \frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{5}{3}+\frac{49}{36}$$

Tāpiri \frac{5}{3} ki te \frac{49}{36} 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.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{109}{36}$$

Tauwehea x^{2}+\frac{7}{3}x+\frac{49}{36}. 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}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{109}{36}$$

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

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{109}{36}}$$

Whakarūnātia.

$$x+\frac{7}{6}=\frac{\sqrt{109}}{6} x+\frac{7}{6}=-\frac{\sqrt{109}}{6}$$

Me tango \frac{7}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{109}-7}{6} x=\frac{-\sqrt{109}-7}{6}$$