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

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

Whakaoti mō x

$$x=\sqrt{53}+7\approx 14.280109889$$

$$x=7-\sqrt{53}\approx -0.280109889$$

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.

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

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

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

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-4\right)}}{2}$$

Whakareatia -4 ki te -4.

$$x=\frac{-\left(-14\right)±\sqrt{196+16}}{2}$$

Tāpiri 196 ki te 16.

$$x=\frac{-\left(-14\right)±\sqrt{212}}{2}$$

Tuhia te pūtakerua o te 212.

$$x=\frac{-\left(-14\right)±2\sqrt{53}}{2}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±2\sqrt{53}}{2}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{53}}{2} ina he tāpiri te ±. Tāpiri 14 ki te 2\sqrt{53}.

$$x=\frac{2\sqrt{53}+14}{2}$$

Whakawehe 14+2\sqrt{53} ki te 2.

$$x=\sqrt{53}+7$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{53}}{2} ina he tango te ±. Tango 2\sqrt{53} mai i 14.

$$x=\frac{14-2\sqrt{53}}{2}$$

Whakawehe 14-2\sqrt{53} ki te 2.

$$x=7-\sqrt{53}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{53}+7 x=7-\sqrt{53}$$

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.

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

Me tāpiri 4 ki ngā taha e rua o te whārite.

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

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

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

Tango -4 mai i 0.

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

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=4+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=4+49$$

Tāpiri 4 ki te 49.

$$x^{2}-14x+49=53$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=53$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{53}$$

Whakarūnātia.

$$x-7=\sqrt{53} x-7=-\sqrt{53}$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=\sqrt{53}+7 x=7-\sqrt{53}$$