Whakaoti i te 6x^2+15x-9=3

$$6 x ^ { 2 } + 15 x - 9 = 3$$

Whakaoti mō x

$$x=\frac{\sqrt{57}-5}{4}\approx 0.637458609$$

$$x=\frac{-\sqrt{57}-5}{4}\approx -3.137458609$$

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.

$$6x^{2}+15x-9=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$6x^{2}+15x-9-3=3-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$6x^{2}+15x-9-3=0$$

Tango 3 mai i -9.

$$6x^{2}+15x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 15 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 6\left(-12\right)}}{2\times 6}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\times 6\left(-12\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-15±\sqrt{225-24\left(-12\right)}}{2\times 6}$$

Whakareatia -24 ki te -12.

$$x=\frac{-15±\sqrt{225+288}}{2\times 6}$$

Tāpiri 225 ki te 288.

$$x=\frac{-15±\sqrt{513}}{2\times 6}$$

Tuhia te pūtakerua o te 513.

$$x=\frac{-15±3\sqrt{57}}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-15±3\sqrt{57}}{12}$$

Nā, me whakaoti te whārite x=\frac{-15±3\sqrt{57}}{12} ina he tāpiri te ±. Tāpiri -15 ki te 3\sqrt{57}.

$$x=\frac{3\sqrt{57}-15}{12}$$

Whakawehe -15+3\sqrt{57} ki te 12.

$$x=\frac{\sqrt{57}-5}{4}$$

Nā, me whakaoti te whārite x=\frac{-15±3\sqrt{57}}{12} ina he tango te ±. Tango 3\sqrt{57} mai i -15.

$$x=\frac{-3\sqrt{57}-15}{12}$$

Whakawehe -15-3\sqrt{57} ki te 12.

$$x=\frac{-\sqrt{57}-5}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{57}-5}{4} x=\frac{-\sqrt{57}-5}{4}$$

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.

$$6x^{2}+15x-9=3$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$6x^{2}+15x-9-\left(-9\right)=3-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$6x^{2}+15x=3-\left(-9\right)$$

Tango -9 mai i 3.

$$6x^{2}+15x=12$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+15x}{6}=\frac{12}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{15}{6}x=\frac{12}{6}$$

Whakahekea te hautanga \frac{15}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

Whakawehe 12 ki te 6.

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

Whakawehea te \frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{4}. Nā, tāpiria te pūrua o te \frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=2+\left(\frac{5}{4}\right)^{2}$$

Pūruatia \frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=2+\frac{25}{16}$$

Tāpiri 2 ki te \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{57}{16}$$

Tauwehea x^{2}+\frac{5}{2}x+\frac{25}{16}. 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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{57}{16}$$

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

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{57}{16}}$$

Whakarūnātia.

$$x+\frac{5}{4}=\frac{\sqrt{57}}{4} x+\frac{5}{4}=-\frac{\sqrt{57}}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{57}-5}{4} x=\frac{-\sqrt{57}-5}{4}$$