Whakaoti i te 7x^2-14x+1/4=0

$$7 x ^ { 2 } - 14 x + \frac { 1 } { 4 } = 0$$

Whakaoti mō x

$$x=\frac{3\sqrt{21}}{14}+1\approx 1.981980506$$

$$x=-\frac{3\sqrt{21}}{14}+1\approx 0.018019494$$

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.

$$7x^{2}-14x+\frac{1}{4}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -14 mō b, me \frac{1}{4} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 7\times \frac{1}{4}}}{2\times 7}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 7\times \frac{1}{4}}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-14\right)±\sqrt{196-28\times \frac{1}{4}}}{2\times 7}$$

Whakareatia -28 ki te \frac{1}{4}.

$$x=\frac{-\left(-14\right)±\sqrt{196-7}}{2\times 7}$$

Tāpiri 196 ki te -7.

$$x=\frac{-\left(-14\right)±\sqrt{189}}{2\times 7}$$

Tuhia te pūtakerua o te 189.

$$x=\frac{-\left(-14\right)±3\sqrt{21}}{2\times 7}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±3\sqrt{21}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{14±3\sqrt{21}}{14}$$

Nā, me whakaoti te whārite x=\frac{14±3\sqrt{21}}{14} ina he tāpiri te ±. Tāpiri 14 ki te 3\sqrt{21}.

$$x=\frac{3\sqrt{21}+14}{14}$$

Whakawehe 14+3\sqrt{21} ki te 14.

$$x=\frac{3\sqrt{21}}{14}+1$$

Nā, me whakaoti te whārite x=\frac{14±3\sqrt{21}}{14} ina he tango te ±. Tango 3\sqrt{21} mai i 14.

$$x=\frac{14-3\sqrt{21}}{14}$$

Whakawehe 14-3\sqrt{21} ki te 14.

$$x=-\frac{3\sqrt{21}}{14}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{3\sqrt{21}}{14}+1 x=-\frac{3\sqrt{21}}{14}+1$$

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.

$$7x^{2}-14x+\frac{1}{4}=0$$

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$7x^{2}-14x+\frac{1}{4}-\frac{1}{4}=-\frac{1}{4}$$

Mā te tango i te \frac{1}{4} i a ia ake anō ka toe ko te 0.

$$7x^{2}-14x=-\frac{1}{4}$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-14x}{7}=-\frac{\frac{1}{4}}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}+\left(-\frac{14}{7}\right)x=-\frac{\frac{1}{4}}{7}$$

Whakawehe -14 ki te 7.

$$x^{2}-2x=-\frac{\frac{1}{4}}{7}$$

Whakawehe -\frac{1}{4} ki te 7.

$$x^{2}-2x=-\frac{1}{28}$$

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.

$$x^{2}-2x+1=-\frac{1}{28}+1$$

Tāpiri -\frac{1}{28} ki te 1.

$$x^{2}-2x+1=\frac{27}{28}$$

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}.

$$\left(x-1\right)^{2}=\frac{27}{28}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{27}{28}}$$

Whakarūnātia.

$$x-1=\frac{3\sqrt{21}}{14} x-1=-\frac{3\sqrt{21}}{14}$$

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

$$x=\frac{3\sqrt{21}}{14}+1 x=-\frac{3\sqrt{21}}{14}+1$$