Whakaoti i te 8*x^2+6x=7

$$8 \cdot x ^ { 2 } + 6 x = 7$$

Whakaoti mō x

$$x=\frac{\sqrt{65}-3}{8}\approx 0.632782219$$

$$x=\frac{-\sqrt{65}-3}{8}\approx -1.382782219$$

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.

$$8x^{2}+6x=7$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$8x^{2}+6x-7=7-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$8x^{2}+6x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 6 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 8\left(-7\right)}}{2\times 8}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 8\left(-7\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-6±\sqrt{36-32\left(-7\right)}}{2\times 8}$$

Whakareatia -32 ki te -7.

$$x=\frac{-6±\sqrt{36+224}}{2\times 8}$$

Tāpiri 36 ki te 224.

$$x=\frac{-6±\sqrt{260}}{2\times 8}$$

Tuhia te pūtakerua o te 260.

$$x=\frac{-6±2\sqrt{65}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-6±2\sqrt{65}}{16}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{65}}{16} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{65}.

$$x=\frac{2\sqrt{65}-6}{16}$$

Whakawehe -6+2\sqrt{65} ki te 16.

$$x=\frac{\sqrt{65}-3}{8}$$

Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{65}}{16} ina he tango te ±. Tango 2\sqrt{65} mai i -6.

$$x=\frac{-2\sqrt{65}-6}{16}$$

Whakawehe -6-2\sqrt{65} ki te 16.

$$x=\frac{-\sqrt{65}-3}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{65}-3}{8} x=\frac{-\sqrt{65}-3}{8}$$

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.

$$8x^{2}+6x=7$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+6x}{8}=\frac{7}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\frac{6}{8}x=\frac{7}{8}$$

Whakahekea te hautanga \frac{6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{3}{4}x=\frac{7}{8}$$

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=\frac{7}{8}+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{7}{8}+\frac{9}{64}$$

Tāpiri \frac{7}{8} ki te \frac{9}{64} 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.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{65}{64}$$

Tauwehea x^{2}+\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x+\frac{3}{8}\right)^{2}=\frac{65}{64}$$

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

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{\frac{65}{64}}$$

Whakarūnātia.

$$x+\frac{3}{8}=\frac{\sqrt{65}}{8} x+\frac{3}{8}=-\frac{\sqrt{65}}{8}$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{65}-3}{8} x=\frac{-\sqrt{65}-3}{8}$$