Whakaoti i te 0.5*x^2+4x-2=0

$$0.5 \times { x }^{ 2 } +4x-2 = 0$$

Whakaoti mō x

$$x=2\sqrt{5}-4\approx 0.472135955$$

$$x=-2\sqrt{5}-4\approx -8.472135955$$

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.

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

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

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

Pūrua 4.

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

Whakareatia -4 ki te \frac{1}{2}.

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

Whakareatia -2 ki te -2.

$$x=\frac{-4±\sqrt{16+4}}{2\times \frac{1}{2}}$$

Tāpiri 16 ki te 4.

$$x=\frac{-4±\sqrt{20}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te 20.

$$x=\frac{-4±2\sqrt{5}}{2\times \frac{1}{2}}$$

Whakareatia 2 ki te \frac{1}{2}.

$$x=\frac{-4±2\sqrt{5}}{1}$$

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{1} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{5}.

$$x=\frac{2\sqrt{5}-4}{1}$$

Whakawehe -4+2\sqrt{5} ki te 1.

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

Nā, me whakaoti te whārite x=\frac{-4±2\sqrt{5}}{1} ina he tango te ±. Tango 2\sqrt{5} mai i -4.

$$x=\frac{-2\sqrt{5}-4}{1}$$

Whakawehe -4-2\sqrt{5} ki te 1.

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

Kua oti te whārite te whakatau.

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

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.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

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

Mā te tango i te -2 i a ia ake anō ka toe ko te 0.

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

Tango -2 mai i 0.

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

Me whakarea ngā taha e rua ki te 2.

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

Mā te whakawehe ki te \frac{1}{2} ka wetekia te whakareanga ki te \frac{1}{2}.

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

Whakawehe 4 ki te \frac{1}{2} mā te whakarea 4 ki te tau huripoki o \frac{1}{2}.

$$x^{2}+8x=\frac{2}{\frac{1}{2}}$$

Whakawehe 2 ki te \frac{1}{2} mā te whakarea 2 ki te tau huripoki o \frac{1}{2}.

$$x^{2}+8x=4$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=4+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=4+16$$

Tāpiri 4 ki te 16.

$$x^{2}+8x+16=20$$

Tauwehea x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=20$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{20}$$

Whakarūnātia.

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

Me tango 4 mai i ngā taha e rua o te whārite.

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