Whakaoti i te 17x^2-6x-15=0

$$17 x ^ { 2 } - 6 x - 15 = 0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{66} + 3}{17} \approx 1.132239812$$

$$x=\frac{3-2\sqrt{66}}{17}\approx -0.779298636$$

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.

$$17x^{2}-6x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 17 mō a, -6 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 17\left(-15\right)}}{2\times 17}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 17\left(-15\right)}}{2\times 17}$$

Whakareatia -4 ki te 17.

$$x=\frac{-\left(-6\right)±\sqrt{36-68\left(-15\right)}}{2\times 17}$$

Whakareatia -68 ki te -15.

$$x=\frac{-\left(-6\right)±\sqrt{36+1020}}{2\times 17}$$

Tāpiri 36 ki te 1020.

$$x=\frac{-\left(-6\right)±\sqrt{1056}}{2\times 17}$$

Tuhia te pūtakerua o te 1056.

$$x=\frac{-\left(-6\right)±4\sqrt{66}}{2\times 17}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±4\sqrt{66}}{2\times 17}$$

Whakareatia 2 ki te 17.

$$x=\frac{6±4\sqrt{66}}{34}$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{66}}{34} ina he tāpiri te ±. Tāpiri 6 ki te 4\sqrt{66}.

$$x=\frac{4\sqrt{66}+6}{34}$$

Whakawehe 6+4\sqrt{66} ki te 34.

$$x=\frac{2\sqrt{66}+3}{17}$$

Nā, me whakaoti te whārite x=\frac{6±4\sqrt{66}}{34} ina he tango te ±. Tango 4\sqrt{66} mai i 6.

$$x=\frac{6-4\sqrt{66}}{34}$$

Whakawehe 6-4\sqrt{66} ki te 34.

$$x=\frac{3-2\sqrt{66}}{17}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{66}+3}{17} x=\frac{3-2\sqrt{66}}{17}$$

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.

$$17x^{2}-6x-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$17x^{2}-6x-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$17x^{2}-6x=-\left(-15\right)$$

Tango -15 mai i 0.

$$17x^{2}-6x=15$$

Whakawehea ngā taha e rua ki te 17.

$$\frac{17x^{2}-6x}{17}=\frac{15}{17}$$

Mā te whakawehe ki te 17 ka wetekia te whakareanga ki te 17.

$$x^{2}-\frac{6}{17}x=\frac{15}{17}$$

Whakawehea te -\frac{6}{17}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{17}. Nā, tāpiria te pūrua o te -\frac{3}{17} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{6}{17}x+\left(-\frac{3}{17}\right)^{2}=\frac{15}{17}+\left(-\frac{3}{17}\right)^{2}$$

Pūruatia -\frac{3}{17} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{15}{17}+\frac{9}{289}$$

Tāpiri \frac{15}{17} ki te \frac{9}{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.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{264}{289}$$

Tauwehea x^{2}-\frac{6}{17}x+\frac{9}{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}.

$$\left(x-\frac{3}{17}\right)^{2}=\frac{264}{289}$$

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

$$\sqrt{\left(x-\frac{3}{17}\right)^{2}}=\sqrt{\frac{264}{289}}$$

Whakarūnātia.

$$x-\frac{3}{17}=\frac{2\sqrt{66}}{17} x-\frac{3}{17}=-\frac{2\sqrt{66}}{17}$$

Me tāpiri \frac{3}{17} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{66}+3}{17} x=\frac{3-2\sqrt{66}}{17}$$