Whakaoti i te a^2-6a-22=0

$$a ^ { 2 } - 6 a - 22 = 0$$

Whakaoti mō a

$$a=\sqrt{31}+3\approx 8.567764363$$

$$a=3-\sqrt{31}\approx -2.567764363$$

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.

$$a^{2}-6a-22=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -22 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-22\right)}}{2}$$

Pūrua -6.

$$a=\frac{-\left(-6\right)±\sqrt{36-4\left(-22\right)}}{2}$$

Whakareatia -4 ki te -22.

$$a=\frac{-\left(-6\right)±\sqrt{36+88}}{2}$$

Tāpiri 36 ki te 88.

$$a=\frac{-\left(-6\right)±\sqrt{124}}{2}$$

Tuhia te pūtakerua o te 124.

$$a=\frac{-\left(-6\right)±2\sqrt{31}}{2}$$

Ko te tauaro o -6 ko 6.

$$a=\frac{6±2\sqrt{31}}{2}$$

Nā, me whakaoti te whārite a=\frac{6±2\sqrt{31}}{2} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{31}.

$$a=\frac{2\sqrt{31}+6}{2}$$

Whakawehe 6+2\sqrt{31} ki te 2.

$$a=\sqrt{31}+3$$

Nā, me whakaoti te whārite a=\frac{6±2\sqrt{31}}{2} ina he tango te ±. Tango 2\sqrt{31} mai i 6.

$$a=\frac{6-2\sqrt{31}}{2}$$

Whakawehe 6-2\sqrt{31} ki te 2.

$$a=3-\sqrt{31}$$

Kua oti te whārite te whakatau.

$$a=\sqrt{31}+3 a=3-\sqrt{31}$$

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.

$$a^{2}-6a-22=0$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$a^{2}-6a-22-\left(-22\right)=-\left(-22\right)$$

Mā te tango i te -22 i a ia ake anō ka toe ko te 0.

$$a^{2}-6a=-\left(-22\right)$$

Tango -22 mai i 0.

$$a^{2}-6a=22$$

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.

$$a^{2}-6a+\left(-3\right)^{2}=22+\left(-3\right)^{2}$$

Pūrua -3.

$$a^{2}-6a+9=22+9$$

Tāpiri 22 ki te 9.

$$a^{2}-6a+9=31$$

Tauwehea a^{2}-6a+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}.

$$\left(a-3\right)^{2}=31$$

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

$$\sqrt{\left(a-3\right)^{2}}=\sqrt{31}$$

Whakarūnātia.

$$a-3=\sqrt{31} a-3=-\sqrt{31}$$

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

$$a=\sqrt{31}+3 a=3-\sqrt{31}$$