Whakaoti i te 4x^2-6=4x

$$4 x ^ { 2 } - 6 = 4 x$$

Whakaoti mō x

$$x = \frac{\sqrt{7} + 1}{2} \approx 1.822875656$$

$$x=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$4x^{2}-6-4x=0$$

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.

$$4x^{2}-4x-6=0$$

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

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

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 4\left(-6\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-4\right)±\sqrt{16-16\left(-6\right)}}{2\times 4}$$

Whakareatia -16 ki te -6.

$$x=\frac{-\left(-4\right)±\sqrt{16+96}}{2\times 4}$$

Tāpiri 16 ki te 96.

$$x=\frac{-\left(-4\right)±\sqrt{112}}{2\times 4}$$

Tuhia te pūtakerua o te 112.

$$x=\frac{-\left(-4\right)±4\sqrt{7}}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4\sqrt{7}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±4\sqrt{7}}{8}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{7}}{8} ina he tāpiri te ±. Tāpiri 4 ki te 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+4}{8}$$

Whakawehe 4+4\sqrt{7} ki te 8.

$$x=\frac{\sqrt{7}+1}{2}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{7}}{8} ina he tango te ±. Tango 4\sqrt{7} mai i 4.

$$x=\frac{4-4\sqrt{7}}{8}$$

Whakawehe 4-4\sqrt{7} ki te 8.

$$x=\frac{1-\sqrt{7}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{7}+1}{2} x=\frac{1-\sqrt{7}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4x mai i ngā taha e rua.

$$4x^{2}-6-4x=0$$

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4x^{2}-4x=6$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{6}{4}$$

Whakawehe -4 ki te 4.

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

Whakahekea te hautanga \frac{6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

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.

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

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

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

Tāpiri \frac{3}{2} 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.

$$x^{2}-x+\frac{1}{4}=\frac{7}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{7}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{7}}{2} x-\frac{1}{2}=-\frac{\sqrt{7}}{2}$$

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

$$x=\frac{\sqrt{7}+1}{2} x=\frac{1-\sqrt{7}}{2}$$