Whakaoti i te 8x^2-24x-24=0

$$8 { x }^{ 2 } -24x-24=0$$

Whakaoti mō x

$$x = \frac{\sqrt{21} + 3}{2} \approx 3.791287847$$

$$x=\frac{3-\sqrt{21}}{2}\approx -0.791287847$$

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.

$$8x^{2}-24x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -24 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 8\left(-24\right)}}{2\times 8}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 8\left(-24\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-24\right)±\sqrt{576-32\left(-24\right)}}{2\times 8}$$

Whakareatia -32 ki te -24.

$$x=\frac{-\left(-24\right)±\sqrt{576+768}}{2\times 8}$$

Tāpiri 576 ki te 768.

$$x=\frac{-\left(-24\right)±\sqrt{1344}}{2\times 8}$$

Tuhia te pūtakerua o te 1344.

$$x=\frac{-\left(-24\right)±8\sqrt{21}}{2\times 8}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±8\sqrt{21}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{24±8\sqrt{21}}{16}$$

Nā, me whakaoti te whārite x=\frac{24±8\sqrt{21}}{16} ina he tāpiri te ±. Tāpiri 24 ki te 8\sqrt{21}.

$$x=\frac{8\sqrt{21}+24}{16}$$

Whakawehe 24+8\sqrt{21} ki te 16.

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

Nā, me whakaoti te whārite x=\frac{24±8\sqrt{21}}{16} ina he tango te ±. Tango 8\sqrt{21} mai i 24.

$$x=\frac{24-8\sqrt{21}}{16}$$

Whakawehe 24-8\sqrt{21} ki te 16.

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

Kua oti te whārite te whakatau.

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

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.

$$8x^{2}-24x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$8x^{2}-24x-24-\left(-24\right)=-\left(-24\right)$$

Mā te tango i te -24 i a ia ake anō ka toe ko te 0.

$$8x^{2}-24x=-\left(-24\right)$$

Tango -24 mai i 0.

$$8x^{2}-24x=24$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-24x}{8}=\frac{24}{8}$$

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

$$x^{2}+\left(-\frac{24}{8}\right)x=\frac{24}{8}$$

Whakawehe -24 ki te 8.

$$x^{2}-3x=\frac{24}{8}$$

Whakawehe 24 ki te 8.

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

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

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=3+\frac{9}{4}$$

Tāpiri 3 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{21}{4}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{21}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

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