Whakaoti i te -x^2-6x+35=0

$$- { x }^{ 2 } -6x+35=0$$

Whakaoti mō x

$$x=2\sqrt{11}-3\approx 3.633249581$$

$$x=-2\sqrt{11}-3\approx -9.633249581$$

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.

$$-x^{2}-6x+35=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -6 mō b, me 35 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 35}}{2\left(-1\right)}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 35}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 35}}{2\left(-1\right)}$$

Whakareatia 4 ki te 35.

$$x=\frac{-\left(-6\right)±\sqrt{36+140}}{2\left(-1\right)}$$

Tāpiri 36 ki te 140.

$$x=\frac{-\left(-6\right)±\sqrt{176}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 176.

$$x=\frac{-\left(-6\right)±4\sqrt{11}}{2\left(-1\right)}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±4\sqrt{11}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{6±4\sqrt{11}}{-2}$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{11}}{-2} ina he tāpiri te ±. Tāpiri 6 ki te 4\sqrt{11}.

$$x=\frac{4\sqrt{11}+6}{-2}$$

Whakawehe 6+4\sqrt{11} ki te -2.

$$x=-2\sqrt{11}-3$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{11}}{-2} ina he tango te ±. Tango 4\sqrt{11} mai i 6.

$$x=\frac{6-4\sqrt{11}}{-2}$$

Whakawehe 6-4\sqrt{11} ki te -2.

$$x=2\sqrt{11}-3$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{11}-3 x=2\sqrt{11}-3$$

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.

$$-x^{2}-6x+35=0$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$-x^{2}-6x+35-35=-35$$

Mā te tango i te 35 i a ia ake anō ka toe ko te 0.

$$-x^{2}-6x=-35$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{35}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{35}{-1}$$

Whakawehe -6 ki te -1.

$$x^{2}+6x=-\frac{35}{-1}$$

Whakawehe -35 ki te -1.

$$x^{2}+6x=35$$

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.

$$x^{2}+6x+3^{2}=35+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=35+9$$

Tāpiri 35 ki te 9.

$$x^{2}+6x+9=44$$

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}.

$$\left(x+3\right)^{2}=44$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{44}$$

Whakarūnātia.

$$x+3=2\sqrt{11} x+3=-2\sqrt{11}$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=2\sqrt{11}-3 x=-2\sqrt{11}-3$$