Whakaoti i te 4x^2-4x-23=0

$$4 x ^ { 2 } - 4 x - 23 = 0$$

Whakaoti mō x

$$x=\sqrt{6}+\frac{1}{2}\approx 2.949489743$$

$$x=\frac{1}{2}-\sqrt{6}\approx -1.949489743$$

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.

$$4x^{2}-4x-23=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -4 mō b, me -23 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 4\left(-23\right)}}{2\times 4}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-23\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-23\right)}}{2\times 4}$$

Whakareatia -16 ki te -23.

$$x=\frac{-\left(-4\right)±\sqrt{16+368}}{2\times 4}$$

Tāpiri 16 ki te 368.

$$x=\frac{-\left(-4\right)±\sqrt{384}}{2\times 4}$$

Tuhia te pūtakerua o te 384.

$$x=\frac{-\left(-4\right)±8\sqrt{6}}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±8\sqrt{6}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±8\sqrt{6}}{8}$$

Nā, me whakaoti te whārite x=\frac{4±8\sqrt{6}}{8} ina he tāpiri te ±. Tāpiri 4 ki te 8\sqrt{6}.

$$x=\frac{8\sqrt{6}+4}{8}$$

Whakawehe 4+8\sqrt{6} ki te 8.

$$x=\sqrt{6}+\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{4±8\sqrt{6}}{8} ina he tango te ±. Tango 8\sqrt{6} mai i 4.

$$x=\frac{4-8\sqrt{6}}{8}$$

Whakawehe 4-8\sqrt{6} ki te 8.

$$x=\frac{1}{2}-\sqrt{6}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{6}+\frac{1}{2} x=\frac{1}{2}-\sqrt{6}$$

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.

$$4x^{2}-4x-23=0$$

Me tāpiri 23 ki ngā taha e rua o te whārite.

$$4x^{2}-4x-23-\left(-23\right)=-\left(-23\right)$$

Mā te tango i te -23 i a ia ake anō ka toe ko te 0.

$$4x^{2}-4x=-\left(-23\right)$$

Tango -23 mai i 0.

$$4x^{2}-4x=23$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-4x}{4}=\frac{23}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{23}{4}$$

Whakawehe -4 ki te 4.

$$x^{2}-x=\frac{23}{4}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{23}{4}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=\frac{23+1}{4}$$

Tāpiri \frac{23}{4} ki te \frac{1}{4} 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.

$$x^{2}-x+\frac{1}{4}=6$$

Tauwehea x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=6$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{6}$$

Whakarūnātia.

$$x-\frac{1}{2}=\sqrt{6} x-\frac{1}{2}=-\sqrt{6}$$

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

$$x=\sqrt{6}+\frac{1}{2} x=\frac{1}{2}-\sqrt{6}$$