Whakaoti i te 25x^2+30x=12

$$25 { x }^{ 2 } +30x=12$$

Whakaoti mō x

$$x=\frac{\sqrt{21}-3}{5}\approx 0.316515139$$

$$x=\frac{-\sqrt{21}-3}{5}\approx -1.516515139$$

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.

$$25x^{2}+30x=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$25x^{2}+30x-12=12-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$25x^{2}+30x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, 30 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 25\left(-12\right)}}{2\times 25}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\times 25\left(-12\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$x=\frac{-30±\sqrt{900-100\left(-12\right)}}{2\times 25}$$

Whakareatia -100 ki te -12.

$$x=\frac{-30±\sqrt{900+1200}}{2\times 25}$$

Tāpiri 900 ki te 1200.

$$x=\frac{-30±\sqrt{2100}}{2\times 25}$$

Tuhia te pūtakerua o te 2100.

$$x=\frac{-30±10\sqrt{21}}{2\times 25}$$

Whakareatia 2 ki te 25.

$$x=\frac{-30±10\sqrt{21}}{50}$$

Nā, me whakaoti te whārite x=\frac{-30±10\sqrt{21}}{50} ina he tāpiri te ±. Tāpiri -30 ki te 10\sqrt{21}.

$$x=\frac{10\sqrt{21}-30}{50}$$

Whakawehe -30+10\sqrt{21} ki te 50.

$$x=\frac{\sqrt{21}-3}{5}$$

Nā, me whakaoti te whārite x=\frac{-30±10\sqrt{21}}{50} ina he tango te ±. Tango 10\sqrt{21} mai i -30.

$$x=\frac{-10\sqrt{21}-30}{50}$$

Whakawehe -30-10\sqrt{21} ki te 50.

$$x=\frac{-\sqrt{21}-3}{5}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{21}-3}{5} x=\frac{-\sqrt{21}-3}{5}$$

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.

$$25x^{2}+30x=12$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25x^{2}+30x}{25}=\frac{12}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$x^{2}+\frac{30}{25}x=\frac{12}{25}$$

Whakahekea te hautanga \frac{30}{25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}+\frac{6}{5}x=\frac{12}{25}$$

Whakawehea te \frac{6}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{5}. Nā, tāpiria te pūrua o te \frac{3}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\frac{12}{25}+\left(\frac{3}{5}\right)^{2}$$

Pūruatia \frac{3}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{12+9}{25}$$

Tāpiri \frac{12}{25} ki te \frac{9}{25} 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.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{21}{25}$$

Tauwehea te x^{2}+\frac{6}{5}x+\frac{9}{25}. 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}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{21}{25}$$

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

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{21}{25}}$$

Whakarūnātia.

$$x+\frac{3}{5}=\frac{\sqrt{21}}{5} x+\frac{3}{5}=-\frac{\sqrt{21}}{5}$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{21}-3}{5} x=\frac{-\sqrt{21}-3}{5}$$