Whakaoti i te 1/3x^2+6x=9

$$\frac { 1 } { 3 } x ^ { 2 } + 6 x = 9$$

Whakaoti mō x

$$x=6\sqrt{3}-9\approx 1.392304845$$

$$x=-6\sqrt{3}-9\approx -19.392304845$$

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.

$$\frac{1}{3}x^{2}+6x=9$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$\frac{1}{3}x^{2}+6x-9=9-9$$

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

$$\frac{1}{3}x^{2}+6x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{3} mō a, 6 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times \frac{1}{3}\left(-9\right)}}{2\times \frac{1}{3}}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times \frac{1}{3}\left(-9\right)}}{2\times \frac{1}{3}}$$

Whakareatia -4 ki te \frac{1}{3}.

$$x=\frac{-6±\sqrt{36-\frac{4}{3}\left(-9\right)}}{2\times \frac{1}{3}}$$

Whakareatia -\frac{4}{3} ki te -9.

$$x=\frac{-6±\sqrt{36+12}}{2\times \frac{1}{3}}$$

Tāpiri 36 ki te 12.

$$x=\frac{-6±\sqrt{48}}{2\times \frac{1}{3}}$$

Tuhia te pūtakerua o te 48.

$$x=\frac{-6±4\sqrt{3}}{2\times \frac{1}{3}}$$

Whakareatia 2 ki te \frac{1}{3}.

$$x=\frac{-6±4\sqrt{3}}{\frac{2}{3}}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{3}}{\frac{2}{3}} ina he tāpiri te ±. Tāpiri -6 ki te 4\sqrt{3}.

$$x=\frac{4\sqrt{3}-6}{\frac{2}{3}}$$

Whakawehe -6+4\sqrt{3} ki te \frac{2}{3} mā te whakarea -6+4\sqrt{3} ki te tau huripoki o \frac{2}{3}.

$$x=6\sqrt{3}-9$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{3}}{\frac{2}{3}} ina he tango te ±. Tango 4\sqrt{3} mai i -6.

$$x=\frac{-4\sqrt{3}-6}{\frac{2}{3}}$$

Whakawehe -6-4\sqrt{3} ki te \frac{2}{3} mā te whakarea -6-4\sqrt{3} ki te tau huripoki o \frac{2}{3}.

$$x=-6\sqrt{3}-9$$

Kua oti te whārite te whakatau.

$$x=6\sqrt{3}-9 x=-6\sqrt{3}-9$$

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.

$$\frac{1}{3}x^{2}+6x=9$$

Me whakarea ngā taha e rua ki te 3.

$$\frac{\frac{1}{3}x^{2}+6x}{\frac{1}{3}}=\frac{9}{\frac{1}{3}}$$

Mā te whakawehe ki te \frac{1}{3} ka wetekia te whakareanga ki te \frac{1}{3}.

$$x^{2}+\frac{6}{\frac{1}{3}}x=\frac{9}{\frac{1}{3}}$$

Whakawehe 6 ki te \frac{1}{3} mā te whakarea 6 ki te tau huripoki o \frac{1}{3}.

$$x^{2}+18x=\frac{9}{\frac{1}{3}}$$

Whakawehe 9 ki te \frac{1}{3} mā te whakarea 9 ki te tau huripoki o \frac{1}{3}.

$$x^{2}+18x=27$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=27+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=27+81$$

Tāpiri 27 ki te 81.

$$x^{2}+18x+81=108$$

Tauwehea te x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=108$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{108}$$

Whakarūnātia.

$$x+9=6\sqrt{3} x+9=-6\sqrt{3}$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=6\sqrt{3}-9 x=-6\sqrt{3}-9$$