Whakaoti i te 0=x^2+4x-1

$$0= { x }^{ 2 } +4x-1$$

Whakaoti mō x (complex solution)

$$x=\sqrt{5}-2\approx 0.236067977$$

$$x=-\left(\sqrt{5}+2\right)\approx -4.236067977$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+4x-1=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)}}{2}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4}}{2}$$

Tāpiri 16 ki te 4.

$$x=\frac{-4±\sqrt{20}}{2}$$

Tuhia te pūtakerua o te 20.

$$x=\frac{-4±2\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{5}.

$$x=\frac{2\sqrt{5}-4}{2}$$

Whakawehe -4+2\sqrt{5} ki te 2.

$$x=\sqrt{5}-2$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{2} ina he tango te ±. Tango 2\sqrt{5} mai i -4.

$$x=\frac{-2\sqrt{5}-4}{2}$$

Whakawehe -4-2\sqrt{5} ki te 2.

$$x=-\sqrt{5}-2$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}-2 x=-\sqrt{5}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+4x-1=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}+4x=1$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=1+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=1+4$$

Tāpiri 1 ki te 4.

$$x^{2}+4x+4=5$$

Tauwehea x^{2}+4x+4. 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}.

$$\left(x+2\right)^{2}=5$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x+2=\sqrt{5} x+2=-\sqrt{5}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{5}-2 x=-\sqrt{5}-2$$

Whakaoti mō x

$$x=\sqrt{5}-2\approx 0.236067977$$

$$x=-\sqrt{5}-2\approx -4.236067977$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+4x-1=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)}}{2}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4}}{2}$$

Tāpiri 16 ki te 4.

$$x=\frac{-4±\sqrt{20}}{2}$$

Tuhia te pūtakerua o te 20.

$$x=\frac{-4±2\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{5}.

$$x=\frac{2\sqrt{5}-4}{2}$$

Whakawehe -4+2\sqrt{5} ki te 2.

$$x=\sqrt{5}-2$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{2} ina he tango te ±. Tango 2\sqrt{5} mai i -4.

$$x=\frac{-2\sqrt{5}-4}{2}$$

Whakawehe -4-2\sqrt{5} ki te 2.

$$x=-\sqrt{5}-2$$

Kua oti te whārite te whakatau.

$$x=\sqrt{5}-2 x=-\sqrt{5}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$x^{2}+4x-1=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}+4x=1$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=1+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=1+4$$

Tāpiri 1 ki te 4.

$$x^{2}+4x+4=5$$

Tauwehea x^{2}+4x+4. 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}.

$$\left(x+2\right)^{2}=5$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$x+2=\sqrt{5} x+2=-\sqrt{5}$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=\sqrt{5}-2 x=-\sqrt{5}-2$$