Whakaoti i te 49x^2-84x+36=0

$$49 x ^ { 2 } - 84 x + 36 = 0$$

Whakaoti mō x

$$x=\frac{6}{7}\approx 0.857142857$$

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.

$$49x^{2}-84x+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, -84 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 49\times 36}}{2\times 49}$$

Pūrua -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 49\times 36}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$x=\frac{-\left(-84\right)±\sqrt{7056-196\times 36}}{2\times 49}$$

Whakareatia -196 ki te 36.

$$x=\frac{-\left(-84\right)±\sqrt{7056-7056}}{2\times 49}$$

Tāpiri 7056 ki te -7056.

$$x=\frac{-\left(-84\right)±\sqrt{0}}{2\times 49}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-84}{2\times 49}$$

Ko te tauaro o -84 ko 84.

$$x=\frac{84}{2\times 49}$$

Whakareatia 2 ki te 49.

$$x=\frac{84}{98}$$

Whakahekea te hautanga \frac{84}{98} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=\frac{6}{7}$$

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.

$$49x^{2}-84x+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$49x^{2}-84x+36-36=-36$$

Mā te tango i te 36 i a ia ake anō ka toe ko te 0.

$$49x^{2}-84x=-36$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49x^{2}-84x}{49}=-\frac{36}{49}$$

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

$$x^{2}+\left(-\frac{84}{49}\right)x=-\frac{36}{49}$$

Whakahekea te hautanga \frac{-84}{49} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$x^{2}-\frac{12}{7}x=-\frac{36}{49}$$

Whakawehea te -\frac{12}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{6}{7}. Nā, tāpiria te pūrua o te -\frac{6}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=-\frac{36}{49}+\left(-\frac{6}{7}\right)^{2}$$

Pūruatia -\frac{6}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{-36+36}{49}$$

Tāpiri -\frac{36}{49} ki te \frac{36}{49} 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.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=0$$

Tauwehea x^{2}-\frac{12}{7}x+\frac{36}{49}. 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}.

$$\left(x-\frac{6}{7}\right)^{2}=0$$

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

$$\sqrt{\left(x-\frac{6}{7}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-\frac{6}{7}=0 x-\frac{6}{7}=0$$

Me tāpiri \frac{6}{7} ki ngā taha e rua o te whārite.

$$x=\frac{6}{7} x=\frac{6}{7}$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=\frac{6}{7}$$