Whakaoti i te 14x^2-64+60x=0

$$14 x ^ { 2 } - 64 + 60 x = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{449}-15}{7}\approx 0.884231443$$

$$x=\frac{-\sqrt{449}-15}{7}\approx -5.169945729$$

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.

$$14x^{2}+60x-64=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 14 mō a, 60 mō b, me -64 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\times 14\left(-64\right)}}{2\times 14}$$

Pūrua 60.

$$x=\frac{-60±\sqrt{3600-4\times 14\left(-64\right)}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$x=\frac{-60±\sqrt{3600-56\left(-64\right)}}{2\times 14}$$

Whakareatia -56 ki te -64.

$$x=\frac{-60±\sqrt{3600+3584}}{2\times 14}$$

Tāpiri 3600 ki te 3584.

$$x=\frac{-60±\sqrt{7184}}{2\times 14}$$

Tuhia te pūtakerua o te 7184.

$$x=\frac{-60±4\sqrt{449}}{2\times 14}$$

Whakareatia 2 ki te 14.

$$x=\frac{-60±4\sqrt{449}}{28}$$

Nā, me whakaoti te whārite x=\frac{-60±4\sqrt{449}}{28} ina he tāpiri te ±. Tāpiri -60 ki te 4\sqrt{449}.

$$x=\frac{4\sqrt{449}-60}{28}$$

Whakawehe -60+4\sqrt{449} ki te 28.

$$x=\frac{\sqrt{449}-15}{7}$$

Nā, me whakaoti te whārite x=\frac{-60±4\sqrt{449}}{28} ina he tango te ±. Tango 4\sqrt{449} mai i -60.

$$x=\frac{-4\sqrt{449}-60}{28}$$

Whakawehe -60-4\sqrt{449} ki te 28.

$$x=\frac{-\sqrt{449}-15}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{449}-15}{7} x=\frac{-\sqrt{449}-15}{7}$$

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.

$$14x^{2}+60x-64=0$$

Me tāpiri 64 ki ngā taha e rua o te whārite.

$$14x^{2}+60x-64-\left(-64\right)=-\left(-64\right)$$

Mā te tango i te -64 i a ia ake anō ka toe ko te 0.

$$14x^{2}+60x=-\left(-64\right)$$

Tango -64 mai i 0.

$$14x^{2}+60x=64$$

Whakawehea ngā taha e rua ki te 14.

$$\frac{14x^{2}+60x}{14}=\frac{64}{14}$$

Mā te whakawehe ki te 14 ka wetekia te whakareanga ki te 14.

$$x^{2}+\frac{60}{14}x=\frac{64}{14}$$

Whakahekea te hautanga \frac{60}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{30}{7}x=\frac{64}{14}$$

Whakahekea te hautanga \frac{64}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{30}{7}x=\frac{32}{7}$$

Whakawehea te \frac{30}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{7}. Nā, tāpiria te pūrua o te \frac{15}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{30}{7}x+\left(\frac{15}{7}\right)^{2}=\frac{32}{7}+\left(\frac{15}{7}\right)^{2}$$

Pūruatia \frac{15}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=\frac{32}{7}+\frac{225}{49}$$

Tāpiri \frac{32}{7} ki te \frac{225}{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.

$$x^{2}+\frac{30}{7}x+\frac{225}{49}=\frac{449}{49}$$

Tauwehea x^{2}+\frac{30}{7}x+\frac{225}{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}.

$$\left(x+\frac{15}{7}\right)^{2}=\frac{449}{49}$$

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

$$\sqrt{\left(x+\frac{15}{7}\right)^{2}}=\sqrt{\frac{449}{49}}$$

Whakarūnātia.

$$x+\frac{15}{7}=\frac{\sqrt{449}}{7} x+\frac{15}{7}=-\frac{\sqrt{449}}{7}$$

Me tango \frac{15}{7} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{449}-15}{7} x=\frac{-\sqrt{449}-15}{7}$$