Whakaoti i te 5x^2-10x-2=0

$$5 x ^ { 2 } - 10 x - 2 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{35}}{5}+1\approx 2.183215957$$

$$x=-\frac{\sqrt{35}}{5}+1\approx -0.183215957$$

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.

$$5x^{2}-10x-2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -10 mō b, me -2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 5\left(-2\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-10\right)±\sqrt{100-20\left(-2\right)}}{2\times 5}$$

Whakareatia -20 ki te -2.

$$x=\frac{-\left(-10\right)±\sqrt{100+40}}{2\times 5}$$

Tāpiri 100 ki te 40.

$$x=\frac{-\left(-10\right)±\sqrt{140}}{2\times 5}$$

Tuhia te pūtakerua o te 140.

$$x=\frac{-\left(-10\right)±2\sqrt{35}}{2\times 5}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±2\sqrt{35}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{10±2\sqrt{35}}{10}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{35}}{10} ina he tāpiri te ±. Tāpiri 10 ki te 2\sqrt{35}.

$$x=\frac{2\sqrt{35}+10}{10}$$

Whakawehe 10+2\sqrt{35} ki te 10.

$$x=\frac{\sqrt{35}}{5}+1$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{35}}{10} ina he tango te ±. Tango 2\sqrt{35} mai i 10.

$$x=\frac{10-2\sqrt{35}}{10}$$

Whakawehe 10-2\sqrt{35} ki te 10.

$$x=-\frac{\sqrt{35}}{5}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{35}}{5}+1 x=-\frac{\sqrt{35}}{5}+1$$

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.

$$5x^{2}-10x-2=0$$

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

$$5x^{2}-10x-2-\left(-2\right)=-\left(-2\right)$$

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

$$5x^{2}-10x=-\left(-2\right)$$

Tango -2 mai i 0.

$$5x^{2}-10x=2$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-10x}{5}=\frac{2}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{10}{5}\right)x=\frac{2}{5}$$

Whakawehe -10 ki te 5.

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

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.

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

Tāpiri \frac{2}{5} ki te 1.

$$x^{2}-2x+1=\frac{7}{5}$$

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}.

$$\left(x-1\right)^{2}=\frac{7}{5}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{7}{5}}$$

Whakarūnātia.

$$x-1=\frac{\sqrt{35}}{5} x-1=-\frac{\sqrt{35}}{5}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{35}}{5}+1 x=-\frac{\sqrt{35}}{5}+1$$