Whakaoti i te 5-6x-x^2=0

$$5 - 6 x - x ^ { 2 } = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{14}-3\approx 0.741657387$$

$$x=-\left(\sqrt{14}+3\right)\approx -6.741657387$$

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.

$$-x^{2}-6x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -6 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 5}}{2\left(-1\right)}$$

Whakareatia 4 ki te 5.

$$x=\frac{-\left(-6\right)±\sqrt{36+20}}{2\left(-1\right)}$$

Tāpiri 36 ki te 20.

$$x=\frac{-\left(-6\right)±\sqrt{56}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 56.

$$x=\frac{-\left(-6\right)±2\sqrt{14}}{2\left(-1\right)}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{14}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{6±2\sqrt{14}}{-2}$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{14}}{-2} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{14}.

$$x=\frac{2\sqrt{14}+6}{-2}$$

Whakawehe 6+2\sqrt{14} ki te -2.

$$x=-\left(\sqrt{14}+3\right)$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{14}}{-2} ina he tango te ±. Tango 2\sqrt{14} mai i 6.

$$x=\frac{6-2\sqrt{14}}{-2}$$

Whakawehe 6-2\sqrt{14} ki te -2.

$$x=\sqrt{14}-3$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{14}+3\right) x=\sqrt{14}-3$$

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.

$$-x^{2}-6x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{5}{-1}$$

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

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{5}{-1}$$

Whakawehe -6 ki te -1.

$$x^{2}+6x=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

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

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.

$$x^{2}+6x+3^{2}=5+3^{2}$$

Pūrua 3.

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

Tāpiri 5 ki te 9.

$$x^{2}+6x+9=14$$

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}.

$$\left(x+3\right)^{2}=14$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{14}$$

Whakarūnātia.

$$x+3=\sqrt{14} x+3=-\sqrt{14}$$

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

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$

Whakaoti mō x

$$x=\sqrt{14}-3\approx 0.741657387$$

$$x=-\sqrt{14}-3\approx -6.741657387$$

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.

$$-x^{2}-6x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -6 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 5}}{2\left(-1\right)}$$

Whakareatia 4 ki te 5.

$$x=\frac{-\left(-6\right)±\sqrt{36+20}}{2\left(-1\right)}$$

Tāpiri 36 ki te 20.

$$x=\frac{-\left(-6\right)±\sqrt{56}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 56.

$$x=\frac{-\left(-6\right)±2\sqrt{14}}{2\left(-1\right)}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{14}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{6±2\sqrt{14}}{-2}$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{14}}{-2} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{14}.

$$x=\frac{2\sqrt{14}+6}{-2}$$

Whakawehe 6+2\sqrt{14} ki te -2.

$$x=-\left(\sqrt{14}+3\right)$$

Nā, me whakaoti te whārite x=\frac{6±2\sqrt{14}}{-2} ina he tango te ±. Tango 2\sqrt{14} mai i 6.

$$x=\frac{6-2\sqrt{14}}{-2}$$

Whakawehe 6-2\sqrt{14} ki te -2.

$$x=\sqrt{14}-3$$

Kua oti te whārite te whakatau.

$$x=-\left(\sqrt{14}+3\right) x=\sqrt{14}-3$$

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.

$$-x^{2}-6x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-6x}{-1}=-\frac{5}{-1}$$

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

$$x^{2}+\left(-\frac{6}{-1}\right)x=-\frac{5}{-1}$$

Whakawehe -6 ki te -1.

$$x^{2}+6x=-\frac{5}{-1}$$

Whakawehe -5 ki te -1.

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

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.

$$x^{2}+6x+3^{2}=5+3^{2}$$

Pūrua 3.

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

Tāpiri 5 ki te 9.

$$x^{2}+6x+9=14$$

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}.

$$\left(x+3\right)^{2}=14$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{14}$$

Whakarūnātia.

$$x+3=\sqrt{14} x+3=-\sqrt{14}$$

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

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$