Whakaoti i te 8x^2+48x+27=0

$$8 x ^ { 2 } + 48 x + 27 = 0$$

Whakaoti mō x

$$x=\frac{3\sqrt{10}}{4}-3\approx -0.628291755$$

$$x=-\frac{3\sqrt{10}}{4}-3\approx -5.371708245$$

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.

$$8x^{2}+48x+27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 48 mō b, me 27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\times 8\times 27}}{2\times 8}$$

Pūrua 48.

$$x=\frac{-48±\sqrt{2304-4\times 8\times 27}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-48±\sqrt{2304-32\times 27}}{2\times 8}$$

Whakareatia -32 ki te 27.

$$x=\frac{-48±\sqrt{2304-864}}{2\times 8}$$

Tāpiri 2304 ki te -864.

$$x=\frac{-48±\sqrt{1440}}{2\times 8}$$

Tuhia te pūtakerua o te 1440.

$$x=\frac{-48±12\sqrt{10}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-48±12\sqrt{10}}{16}$$

Nā, me whakaoti te whārite x=\frac{-48±12\sqrt{10}}{16} ina he tāpiri te ±. Tāpiri -48 ki te 12\sqrt{10}.

$$x=\frac{12\sqrt{10}-48}{16}$$

Whakawehe -48+12\sqrt{10} ki te 16.

$$x=\frac{3\sqrt{10}}{4}-3$$

Nā, me whakaoti te whārite x=\frac{-48±12\sqrt{10}}{16} ina he tango te ±. Tango 12\sqrt{10} mai i -48.

$$x=\frac{-12\sqrt{10}-48}{16}$$

Whakawehe -48-12\sqrt{10} ki te 16.

$$x=-\frac{3\sqrt{10}}{4}-3$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{10}}{4}-3 x=-\frac{3\sqrt{10}}{4}-3$$

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.

$$8x^{2}+48x+27=0$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$8x^{2}+48x+27-27=-27$$

Mā te tango i te 27 i a ia ake anō ka toe ko te 0.

$$8x^{2}+48x=-27$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+48x}{8}=-\frac{27}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{48}{8}x=-\frac{27}{8}$$

Whakawehe 48 ki te 8.

$$x^{2}+6x=-\frac{27}{8}$$

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

$$x^{2}+6x+3^{2}=-\frac{27}{8}+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-\frac{27}{8}+9$$

Tāpiri -\frac{27}{8} ki te 9.

$$x^{2}+6x+9=\frac{45}{8}$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=\frac{45}{8}$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{\frac{45}{8}}$$

Whakarūnātia.

$$x+3=\frac{3\sqrt{10}}{4} x+3=-\frac{3\sqrt{10}}{4}$$

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

$$x=\frac{3\sqrt{10}}{4}-3 x=-\frac{3\sqrt{10}}{4}-3$$