Whakaoti i te 10x^2-15x+2=0

$$10 x ^ { 2 } - 15 x + 2 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4}\approx 1.352079729$$

$$x=-\frac{\sqrt{145}}{20}+\frac{3}{4}\approx 0.147920271$$

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.

$$10x^{2}-15x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -15 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 10\times 2}}{2\times 10}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 10\times 2}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-15\right)±\sqrt{225-40\times 2}}{2\times 10}$$

Whakareatia -40 ki te 2.

$$x=\frac{-\left(-15\right)±\sqrt{225-80}}{2\times 10}$$

Tāpiri 225 ki te -80.

$$x=\frac{-\left(-15\right)±\sqrt{145}}{2\times 10}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±\sqrt{145}}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{15±\sqrt{145}}{20}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{145}}{20} ina he tāpiri te ±. Tāpiri 15 ki te \sqrt{145}.

$$x=\frac{\sqrt{145}+15}{20}$$

Whakawehe 15+\sqrt{145} ki te 20.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4}$$

Nā, me whakaoti te whārite x=\frac{15±\sqrt{145}}{20} ina he tango te ±. Tango \sqrt{145} mai i 15.

$$x=\frac{15-\sqrt{145}}{20}$$

Whakawehe 15-\sqrt{145} ki te 20.

$$x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4} x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$

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.

$$10x^{2}-15x+2=0$$

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

$$10x^{2}-15x+2-2=-2$$

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

$$10x^{2}-15x=-2$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-15x}{10}=-\frac{2}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\left(-\frac{15}{10}\right)x=-\frac{2}{10}$$

Whakahekea te hautanga \frac{-15}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{3}{2}x=-\frac{2}{10}$$

Whakahekea te hautanga \frac{-2}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{3}{2}x=-\frac{1}{5}$$

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{1}{5}+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{5}+\frac{9}{16}$$

Tāpiri -\frac{1}{5} ki te \frac{9}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{29}{80}$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{29}{80}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{29}{80}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{\sqrt{145}}{20} x-\frac{3}{4}=-\frac{\sqrt{145}}{20}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4} x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$