Whakaoti i te x^2-4x-14=0

$$x ^ { 2 } - 4 x - 14 = 0$$

Whakaoti mō x

$$x=3\sqrt{2}+2\approx 6.242640687$$

$$x=2-3\sqrt{2}\approx -2.242640687$$

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.

$$x^{2}-4x-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-14\right)}}{2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-14\right)}}{2}$$

Whakareatia -4 ki te -14.

$$x=\frac{-\left(-4\right)±\sqrt{16+56}}{2}$$

Tāpiri 16 ki te 56.

$$x=\frac{-\left(-4\right)±\sqrt{72}}{2}$$

Tuhia te pūtakerua o te 72.

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

Ko te tauaro o -4 ko 4.

$$x=\frac{4±6\sqrt{2}}{2}$$

Nā, me whakaoti te whārite x=\frac{4±6\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri 4 ki te 6\sqrt{2}.

$$x=\frac{6\sqrt{2}+4}{2}$$

Whakawehe 4+6\sqrt{2} ki te 2.

$$x=3\sqrt{2}+2$$

Nā, me whakaoti te whārite x=\frac{4±6\sqrt{2}}{2} ina he tango te ±. Tango 6\sqrt{2} mai i 4.

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

Whakawehe 4-6\sqrt{2} ki te 2.

$$x=2-3\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{2}+2 x=2-3\sqrt{2}$$

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.

$$x^{2}-4x-14=0$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$x^{2}-4x-14-\left(-14\right)=-\left(-14\right)$$

Mā te tango i te -14 i a ia ake anō ka toe ko te 0.

$$x^{2}-4x=-\left(-14\right)$$

Tango -14 mai i 0.

$$x^{2}-4x=14$$

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

$$x^{2}-4x+\left(-2\right)^{2}=14+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=14+4$$

Tāpiri 14 ki te 4.

$$x^{2}-4x+4=18$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=18$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{18}$$

Whakarūnātia.

$$x-2=3\sqrt{2} x-2=-3\sqrt{2}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=3\sqrt{2}+2 x=2-3\sqrt{2}$$