Whakaoti i te 3x^2+6x-25=0

$$3 { x }^{ 2 } +6x-25=0$$

Whakaoti mō x

$$x=\frac{2\sqrt{21}}{3}-1\approx 2.055050463$$

$$x=-\frac{2\sqrt{21}}{3}-1\approx -4.055050463$$

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.

$$3x^{2}+6x-25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 6 mō b, me -25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\left(-25\right)}}{2\times 3}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-25\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-6±\sqrt{36-12\left(-25\right)}}{2\times 3}$$

Whakareatia -12 ki te -25.

$$x=\frac{-6±\sqrt{36+300}}{2\times 3}$$

Tāpiri 36 ki te 300.

$$x=\frac{-6±\sqrt{336}}{2\times 3}$$

Tuhia te pūtakerua o te 336.

$$x=\frac{-6±4\sqrt{21}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-6±4\sqrt{21}}{6}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{21}}{6} ina he tāpiri te ±. Tāpiri -6 ki te 4\sqrt{21}.

$$x=\frac{4\sqrt{21}-6}{6}$$

Whakawehe -6+4\sqrt{21} ki te 6.

$$x=\frac{2\sqrt{21}}{3}-1$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{21}}{6} ina he tango te ±. Tango 4\sqrt{21} mai i -6.

$$x=\frac{-4\sqrt{21}-6}{6}$$

Whakawehe -6-4\sqrt{21} ki te 6.

$$x=-\frac{2\sqrt{21}}{3}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{21}}{3}-1 x=-\frac{2\sqrt{21}}{3}-1$$

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.

$$3x^{2}+6x-25=0$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

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

Mā te tango i te -25 i a ia ake anō ka toe ko te 0.

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

Tango -25 mai i 0.

$$3x^{2}+6x=25$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+6x}{3}=\frac{25}{3}$$

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

$$x^{2}+\frac{6}{3}x=\frac{25}{3}$$

Whakawehe 6 ki te 3.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=\frac{25}{3}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{25}{3}+1$$

Tāpiri \frac{25}{3} ki te 1.

$$x^{2}+2x+1=\frac{28}{3}$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=\frac{28}{3}$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{28}{3}}$$

Whakarūnātia.

$$x+1=\frac{2\sqrt{21}}{3} x+1=-\frac{2\sqrt{21}}{3}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{21}}{3}-1 x=-\frac{2\sqrt{21}}{3}-1$$