Whakaoti i te x^2-6/8x-1/2=0

$${ x }^{ 2 } - \frac{ 6 }{ 8 } x- \frac{ 1 }{ 2 } =0$$

Whakaoti mō x

$$x = \frac{\sqrt{41} + 3}{8} \approx 1.17539053$$

$$x=\frac{3-\sqrt{41}}{8}\approx -0.42539053$$

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.

$$x^{2}-\frac{3}{4}x-\frac{1}{2}=0$$

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

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

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-4\left(-\frac{1}{2}\right)}}{2}$$

Whakareatia -4 ki te -\frac{1}{2}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+2}}{2}$$

Tāpiri \frac{9}{16} ki te 2.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{41}{16}}}{2}$$

Tuhia te pūtakerua o te \frac{41}{16}.

$$x=\frac{-\left(-\frac{3}{4}\right)±\frac{\sqrt{41}}{4}}{2}$$

Ko te tauaro o -\frac{3}{4} ko \frac{3}{4}.

$$x=\frac{\frac{3}{4}±\frac{\sqrt{41}}{4}}{2}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{4}±\frac{\sqrt{41}}{4}}{2} ina he tāpiri te ±. Tāpiri \frac{3}{4} ki te \frac{\sqrt{41}}{4}.

$$x=\frac{\sqrt{41}+3}{2\times 4}$$

Whakawehe \frac{3+\sqrt{41}}{4} ki te 2.

$$x=\frac{\sqrt{41}+3}{8}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{4}±\frac{\sqrt{41}}{4}}{2} ina he tango te ±. Tango \frac{\sqrt{41}}{4} mai i \frac{3}{4}.

$$x=\frac{3-\sqrt{41}}{2\times 4}$$

Whakawehe \frac{3-\sqrt{41}}{4} ki te 2.

$$x=\frac{3-\sqrt{41}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{41}+3}{8} x=\frac{3-\sqrt{41}}{8}$$

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.

$$x^{2}-\frac{3}{4}x-\frac{1}{2}=0$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x^{2}-\frac{3}{4}x-\frac{1}{2}-\left(-\frac{1}{2}\right)=-\left(-\frac{1}{2}\right)$$

Mā te tango i te -\frac{1}{2} i a ia ake anō ka toe ko te 0.

$$x^{2}-\frac{3}{4}x=-\left(-\frac{1}{2}\right)$$

Tango -\frac{1}{2} mai i 0.

$$x^{2}-\frac{3}{4}x=\frac{1}{2}$$

Whakawehea te -\frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{8}. Nā, tāpiria te pūrua o te -\frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{1}{2}+\left(-\frac{3}{8}\right)^{2}$$

Pūruatia -\frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{1}{2}+\frac{9}{64}$$

Tāpiri \frac{1}{2} ki te \frac{9}{64} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{41}{64}$$

Tauwehea x^{2}-\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{41}{64}$$

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

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{41}{64}}$$

Whakarūnātia.

$$x-\frac{3}{8}=\frac{\sqrt{41}}{8} x-\frac{3}{8}=-\frac{\sqrt{41}}{8}$$

Me tāpiri \frac{3}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{41}+3}{8} x=\frac{3-\sqrt{41}}{8}$$