Whakaoti i te 17/15x^2-4x+1=0

$$\frac { 17 } { 15 } x ^ { 2 } - 4 x + 1 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{645} + 30}{17} \approx 3.258638247$$

$$x=\frac{30-\sqrt{645}}{17}\approx 0.270773518$$

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.

$$\frac{17}{15}x^{2}-4x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{17}{15} mō a, -4 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times \frac{17}{15}}}{2\times \frac{17}{15}}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times \frac{17}{15}}}{2\times \frac{17}{15}}$$

Whakareatia -4 ki te \frac{17}{15}.

$$x=\frac{-\left(-4\right)±\sqrt{16-\frac{68}{15}}}{2\times \frac{17}{15}}$$

Tāpiri 16 ki te -\frac{68}{15}.

$$x=\frac{-\left(-4\right)±\sqrt{\frac{172}{15}}}{2\times \frac{17}{15}}$$

Tuhia te pūtakerua o te \frac{172}{15}.

$$x=\frac{-\left(-4\right)±\frac{2\sqrt{645}}{15}}{2\times \frac{17}{15}}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±\frac{2\sqrt{645}}{15}}{2\times \frac{17}{15}}$$

Whakareatia 2 ki te \frac{17}{15}.

$$x=\frac{4±\frac{2\sqrt{645}}{15}}{\frac{34}{15}}$$

Nā, me whakaoti te whārite x=\frac{4±\frac{2\sqrt{645}}{15}}{\frac{34}{15}} ina he tāpiri te ±. Tāpiri 4 ki te \frac{2\sqrt{645}}{15}.

$$x=\frac{\frac{2\sqrt{645}}{15}+4}{\frac{34}{15}}$$

Whakawehe 4+\frac{2\sqrt{645}}{15} ki te \frac{34}{15} mā te whakarea 4+\frac{2\sqrt{645}}{15} ki te tau huripoki o \frac{34}{15}.

$$x=\frac{\sqrt{645}+30}{17}$$

Nā, me whakaoti te whārite x=\frac{4±\frac{2\sqrt{645}}{15}}{\frac{34}{15}} ina he tango te ±. Tango \frac{2\sqrt{645}}{15} mai i 4.

$$x=\frac{-\frac{2\sqrt{645}}{15}+4}{\frac{34}{15}}$$

Whakawehe 4-\frac{2\sqrt{645}}{15} ki te \frac{34}{15} mā te whakarea 4-\frac{2\sqrt{645}}{15} ki te tau huripoki o \frac{34}{15}.

$$x=\frac{30-\sqrt{645}}{17}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{645}+30}{17} x=\frac{30-\sqrt{645}}{17}$$

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.

$$\frac{17}{15}x^{2}-4x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$\frac{17}{15}x^{2}-4x+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$\frac{17}{15}x^{2}-4x=-1$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{15}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{17}{15}x^{2}-4x}{\frac{17}{15}}=-\frac{1}{\frac{17}{15}}$$

Mā te whakawehe ki te \frac{17}{15} ka wetekia te whakareanga ki te \frac{17}{15}.

$$x^{2}+\left(-\frac{4}{\frac{17}{15}}\right)x=-\frac{1}{\frac{17}{15}}$$

Whakawehe -4 ki te \frac{17}{15} mā te whakarea -4 ki te tau huripoki o \frac{17}{15}.

$$x^{2}-\frac{60}{17}x=-\frac{1}{\frac{17}{15}}$$

Whakawehe -1 ki te \frac{17}{15} mā te whakarea -1 ki te tau huripoki o \frac{17}{15}.

$$x^{2}-\frac{60}{17}x=-\frac{15}{17}$$

Whakawehea te -\frac{60}{17}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{30}{17}. Nā, tāpiria te pūrua o te -\frac{30}{17} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{60}{17}x+\left(-\frac{30}{17}\right)^{2}=-\frac{15}{17}+\left(-\frac{30}{17}\right)^{2}$$

Pūruatia -\frac{30}{17} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{60}{17}x+\frac{900}{289}=-\frac{15}{17}+\frac{900}{289}$$

Tāpiri -\frac{15}{17} ki te \frac{900}{289} 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.

$$x^{2}-\frac{60}{17}x+\frac{900}{289}=\frac{645}{289}$$

Tauwehea x^{2}-\frac{60}{17}x+\frac{900}{289}. 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}.

$$\left(x-\frac{30}{17}\right)^{2}=\frac{645}{289}$$

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

$$\sqrt{\left(x-\frac{30}{17}\right)^{2}}=\sqrt{\frac{645}{289}}$$

Whakarūnātia.

$$x-\frac{30}{17}=\frac{\sqrt{645}}{17} x-\frac{30}{17}=-\frac{\sqrt{645}}{17}$$

Me tāpiri \frac{30}{17} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{645}+30}{17} x=\frac{30-\sqrt{645}}{17}$$