Whakaoti i te x^2+12x-25=0

$${ x }^{ 2 } +12x-25=0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{61}-6\approx 1.810249676$$

$$x=-\left(\sqrt{61}+6\right)\approx -13.810249676$$

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.

$$x^{2}+12x-25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 12 mō b, me -25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-25\right)}}{2}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-25\right)}}{2}$$

Whakareatia -4 ki te -25.

$$x=\frac{-12±\sqrt{144+100}}{2}$$

Tāpiri 144 ki te 100.

$$x=\frac{-12±\sqrt{244}}{2}$$

Tuhia te pūtakerua o te 244.

$$x=\frac{-12±2\sqrt{61}}{2}$$

Nā, me whakaoti te whārite x=\frac{-12±2\sqrt{61}}{2} ina he tāpiri te ±. Tāpiri -12 ki te 2\sqrt{61}.

$$x=\frac{2\sqrt{61}-12}{2}$$

Whakawehe -12+2\sqrt{61} ki te 2.

$$x=\sqrt{61}-6$$

Nā, me whakaoti te whārite x=\frac{-12±2\sqrt{61}}{2} ina he tango te ±. Tango 2\sqrt{61} mai i -12.

$$x=\frac{-2\sqrt{61}-12}{2}$$

Whakawehe -12-2\sqrt{61} ki te 2.

$$x=-\sqrt{61}-6$$

Kua oti te whārite te whakatau.

$$x=\sqrt{61}-6 x=-\sqrt{61}-6$$

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.

$$x^{2}+12x-25=0$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$x^{2}+12x-25-\left(-25\right)=-\left(-25\right)$$

Mā te tango i te -25 i a ia ake anō ka toe ko te 0.

$$x^{2}+12x=-\left(-25\right)$$

Tango -25 mai i 0.

$$x^{2}+12x=25$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=25+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=25+36$$

Tāpiri 25 ki te 36.

$$x^{2}+12x+36=61$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=61$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{61}$$

Whakarūnātia.

$$x+6=\sqrt{61} x+6=-\sqrt{61}$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=\sqrt{61}-6 x=-\sqrt{61}-6$$

Whakaoti mō x

$$x=\sqrt{61}-6\approx 1.810249676$$

$$x=-\sqrt{61}-6\approx -13.810249676$$

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.

$$x^{2}+12x-25=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 12 mō b, me -25 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-25\right)}}{2}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-25\right)}}{2}$$

Whakareatia -4 ki te -25.

$$x=\frac{-12±\sqrt{144+100}}{2}$$

Tāpiri 144 ki te 100.

$$x=\frac{-12±\sqrt{244}}{2}$$

Tuhia te pūtakerua o te 244.

$$x=\frac{-12±2\sqrt{61}}{2}$$

Nā, me whakaoti te whārite x=\frac{-12±2\sqrt{61}}{2} ina he tāpiri te ±. Tāpiri -12 ki te 2\sqrt{61}.

$$x=\frac{2\sqrt{61}-12}{2}$$

Whakawehe -12+2\sqrt{61} ki te 2.

$$x=\sqrt{61}-6$$

Nā, me whakaoti te whārite x=\frac{-12±2\sqrt{61}}{2} ina he tango te ±. Tango 2\sqrt{61} mai i -12.

$$x=\frac{-2\sqrt{61}-12}{2}$$

Whakawehe -12-2\sqrt{61} ki te 2.

$$x=-\sqrt{61}-6$$

Kua oti te whārite te whakatau.

$$x=\sqrt{61}-6 x=-\sqrt{61}-6$$

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.

$$x^{2}+12x-25=0$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$x^{2}+12x-25-\left(-25\right)=-\left(-25\right)$$

Mā te tango i te -25 i a ia ake anō ka toe ko te 0.

$$x^{2}+12x=-\left(-25\right)$$

Tango -25 mai i 0.

$$x^{2}+12x=25$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=25+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=25+36$$

Tāpiri 25 ki te 36.

$$x^{2}+12x+36=61$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=61$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{61}$$

Whakarūnātia.

$$x+6=\sqrt{61} x+6=-\sqrt{61}$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=\sqrt{61}-6 x=-\sqrt{61}-6$$