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

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

Whakaoti mō x

$$x = \frac{\sqrt{37} + 5}{2} \approx 5.541381265$$

$$x=\frac{5-\sqrt{37}}{2}\approx -0.541381265$$

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.

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

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

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

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

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

Tango 3 mai i -6.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -15 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\left(-9\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\left(-9\right)}}{2\times 3}$$

Whakareatia -12 ki te -9.

$$x=\frac{-\left(-15\right)±\sqrt{225+108}}{2\times 3}$$

Tāpiri 225 ki te 108.

$$x=\frac{-\left(-15\right)±\sqrt{333}}{2\times 3}$$

Tuhia te pūtakerua o te 333.

$$x=\frac{-\left(-15\right)±3\sqrt{37}}{2\times 3}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±3\sqrt{37}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{15±3\sqrt{37}}{6}$$

Nā, me whakaoti te whārite x=\frac{15±3\sqrt{37}}{6} ina he tāpiri te ±. Tāpiri 15 ki te 3\sqrt{37}.

$$x=\frac{3\sqrt{37}+15}{6}$$

Whakawehe 15+3\sqrt{37} ki te 6.

$$x=\frac{\sqrt{37}+5}{2}$$

Nā, me whakaoti te whārite x=\frac{15±3\sqrt{37}}{6} ina he tango te ±. Tango 3\sqrt{37} mai i 15.

$$x=\frac{15-3\sqrt{37}}{6}$$

Whakawehe 15-3\sqrt{37} ki te 6.

$$x=\frac{5-\sqrt{37}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{37}+5}{2} x=\frac{5-\sqrt{37}}{2}$$

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.

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

Me tāpiri 6 ki ngā taha e rua o te whārite.

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

Mā te tango i te -6 i a ia ake anō ka toe ko te 0.

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

Tango -6 mai i 3.

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

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-15x}{3}=\frac{9}{3}$$

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

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

Whakawehe -15 ki te 3.

$$x^{2}-5x=\frac{9}{3}$$

Whakawehe 9 ki te 3.

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

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

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

Pūruatia -\frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-5x+\frac{25}{4}=3+\frac{25}{4}$$

Tāpiri 3 ki te \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{37}{4}$$

Tauwehea x^{2}-5x+\frac{25}{4}. 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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{37}{4}$$

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

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Whakarūnātia.

$$x-\frac{5}{2}=\frac{\sqrt{37}}{2} x-\frac{5}{2}=-\frac{\sqrt{37}}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{37}+5}{2} x=\frac{5-\sqrt{37}}{2}$$