Whakaoti i te 4x^2-4x-16=0

$$4 { x }^{ 2 } -4x-16=0$$

Whakaoti mō x

$$x = \frac{\sqrt{17} + 1}{2} \approx 2.561552813$$

$$x=\frac{1-\sqrt{17}}{2}\approx -1.561552813$$

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.

$$4x^{2}-4x-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -4 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -4.

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

Whakareatia -4 ki te 4.

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

Whakareatia -16 ki te -16.

$$x=\frac{-\left(-4\right)±\sqrt{16+256}}{2\times 4}$$

Tāpiri 16 ki te 256.

$$x=\frac{-\left(-4\right)±\sqrt{272}}{2\times 4}$$

Tuhia te pūtakerua o te 272.

$$x=\frac{-\left(-4\right)±4\sqrt{17}}{2\times 4}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±4\sqrt{17}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{4±4\sqrt{17}}{8}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{17}}{8} ina he tāpiri te ±. Tāpiri 4 ki te 4\sqrt{17}.

$$x=\frac{4\sqrt{17}+4}{8}$$

Whakawehe 4+4\sqrt{17} ki te 8.

$$x=\frac{\sqrt{17}+1}{2}$$

Nā, me whakaoti te whārite x=\frac{4±4\sqrt{17}}{8} ina he tango te ±. Tango 4\sqrt{17} mai i 4.

$$x=\frac{4-4\sqrt{17}}{8}$$

Whakawehe 4-4\sqrt{17} ki te 8.

$$x=\frac{1-\sqrt{17}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$

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.

$$4x^{2}-4x-16=0$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$4x^{2}-4x-16-\left(-16\right)=-\left(-16\right)$$

Mā te tango i te -16 i a ia ake anō ka toe ko te 0.

$$4x^{2}-4x=-\left(-16\right)$$

Tango -16 mai i 0.

$$4x^{2}-4x=16$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-4x}{4}=\frac{16}{4}$$

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

$$x^{2}+\left(-\frac{4}{4}\right)x=\frac{16}{4}$$

Whakawehe -4 ki te 4.

$$x^{2}-x=\frac{16}{4}$$

Whakawehe 16 ki te 4.

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

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.

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

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

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

Tāpiri 4 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{17}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{17}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{17}}{2} x-\frac{1}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$