Whakaoti i te 12x^2+25x-45=0

$$12 { x }^{ 2 } +25x-45=0$$

Whakaoti mō x

$$x = \frac{\sqrt{2785} - 25}{24} \approx 1.157212467$$

$$x=\frac{-\sqrt{2785}-25}{24}\approx -3.2405458$$

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.

$$12x^{2}+25x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, 25 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 12\left(-45\right)}}{2\times 12}$$

Pūrua 25.

$$x=\frac{-25±\sqrt{625-4\times 12\left(-45\right)}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-25±\sqrt{625-48\left(-45\right)}}{2\times 12}$$

Whakareatia -48 ki te -45.

$$x=\frac{-25±\sqrt{625+2160}}{2\times 12}$$

Tāpiri 625 ki te 2160.

$$x=\frac{-25±\sqrt{2785}}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-25±\sqrt{2785}}{24}$$

Nā, me whakaoti te whārite x=\frac{-25±\sqrt{2785}}{24} ina he tāpiri te ±. Tāpiri -25 ki te \sqrt{2785}.

$$x=\frac{\sqrt{2785}-25}{24}$$

Nā, me whakaoti te whārite x=\frac{-25±\sqrt{2785}}{24} ina he tango te ±. Tango \sqrt{2785} mai i -25.

$$x=\frac{-\sqrt{2785}-25}{24}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{2785}-25}{24} x=\frac{-\sqrt{2785}-25}{24}$$

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.

$$12x^{2}+25x-45=0$$

Me tāpiri 45 ki ngā taha e rua o te whārite.

$$12x^{2}+25x-45-\left(-45\right)=-\left(-45\right)$$

Mā te tango i te -45 i a ia ake anō ka toe ko te 0.

$$12x^{2}+25x=-\left(-45\right)$$

Tango -45 mai i 0.

$$12x^{2}+25x=45$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}+25x}{12}=\frac{45}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\frac{25}{12}x=\frac{45}{12}$$

Whakahekea te hautanga \frac{45}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{25}{12}x=\frac{15}{4}$$

Whakawehea te \frac{25}{12}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{25}{24}. Nā, tāpiria te pūrua o te \frac{25}{24} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{25}{12}x+\left(\frac{25}{24}\right)^{2}=\frac{15}{4}+\left(\frac{25}{24}\right)^{2}$$

Pūruatia \frac{25}{24} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{25}{12}x+\frac{625}{576}=\frac{15}{4}+\frac{625}{576}$$

Tāpiri \frac{15}{4} ki te \frac{625}{576} 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.

$$x^{2}+\frac{25}{12}x+\frac{625}{576}=\frac{2785}{576}$$

Tauwehea te x^{2}+\frac{25}{12}x+\frac{625}{576}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{24}\right)^{2}=\frac{2785}{576}$$

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

$$\sqrt{\left(x+\frac{25}{24}\right)^{2}}=\sqrt{\frac{2785}{576}}$$

Whakarūnātia.

$$x+\frac{25}{24}=\frac{\sqrt{2785}}{24} x+\frac{25}{24}=-\frac{\sqrt{2785}}{24}$$

Me tango \frac{25}{24} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{2785}-25}{24} x=\frac{-\sqrt{2785}-25}{24}$$