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

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

Whakaoti mō x

$$x=\frac{\sqrt{2}}{2}-1\approx -0.292893219$$

$$x=-\frac{\sqrt{2}}{2}-1\approx -1.707106781$$

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.

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

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

$$x=\frac{-4±\sqrt{4^{2}-4\times 2}}{2\times 2}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 2}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-4±\sqrt{16-8}}{2\times 2}$$

Tāpiri 16 ki te -8.

$$x=\frac{-4±\sqrt{8}}{2\times 2}$$

Tuhia te pūtakerua o te 8.

$$x=\frac{-4±2\sqrt{2}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-4±2\sqrt{2}}{4}$$

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

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

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

$$x=\frac{\sqrt{2}}{2}-1$$

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

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

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

$$x=-\frac{\sqrt{2}}{2}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{2}}{2}-1 x=-\frac{\sqrt{2}}{2}-1$$

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.

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

Me tango 1 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+4x}{2}=-\frac{1}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{4}{2}x=-\frac{1}{2}$$

Whakawehe 4 ki te 2.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūrua 1.

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

Tāpiri -\frac{1}{2} ki te 1.

$$x^{2}+2x+1=\frac{1}{2}$$

Tauwehea x^{2}+2x+1. 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}.

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

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{1}{2}}$$

Whakarūnātia.

$$x+1=\frac{\sqrt{2}}{2} x+1=-\frac{\sqrt{2}}{2}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{2}}{2}-1 x=-\frac{\sqrt{2}}{2}-1$$