Whakaoti i te x^2-6x+4=0

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

Whakaoti mō x

$$x=\sqrt{5}+3\approx 5.236067977$$

$$x=3-\sqrt{5}\approx 0.763932023$$

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 4}}{2}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-6\right)±\sqrt{36-16}}{2}$$

Tāpiri 36 ki te -16.

$$x=\frac{-\left(-6\right)±\sqrt{20}}{2}$$

Tuhia te pūtakerua o te 20.

$$x=\frac{-\left(-6\right)±2\sqrt{5}}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{5}.

$$x=\frac{2\sqrt{5}+6}{2}$$

Whakawehe 6+2\sqrt{5} ki te 2.

$$x=\sqrt{5}+3$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{5}}{2} ina he tango te ±. Tango 2\sqrt{5} mai i 6.

$$x=\frac{6-2\sqrt{5}}{2}$$

Whakawehe 6-2\sqrt{5} ki te 2.

$$x=3-\sqrt{5}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}+3 x=3-\sqrt{5}$$

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.

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

Me tango 4 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

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

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-4+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-4+9$$

Tāpiri -4 ki te 9.

$$x^{2}-6x+9=5$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=5$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x-3=\sqrt{5} x-3=-\sqrt{5}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=\sqrt{5}+3 x=3-\sqrt{5}$$