Whakaoti i te 7875x^2+1425x-1=0

$$7875 { x }^{ 2 } +1425x-1=0$$

Whakaoti mō x

$$x=\frac{\sqrt{9165}}{1050}-\frac{19}{210}\approx 0.000699054$$

$$x=-\frac{\sqrt{9165}}{1050}-\frac{19}{210}\approx -0.181651435$$

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.

$$7875x^{2}+1425x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7875 mō a, 1425 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1425±\sqrt{1425^{2}-4\times 7875\left(-1\right)}}{2\times 7875}$$

Pūrua 1425.

$$x=\frac{-1425±\sqrt{2030625-4\times 7875\left(-1\right)}}{2\times 7875}$$

Whakareatia -4 ki te 7875.

$$x=\frac{-1425±\sqrt{2030625-31500\left(-1\right)}}{2\times 7875}$$

Whakareatia -31500 ki te -1.

$$x=\frac{-1425±\sqrt{2030625+31500}}{2\times 7875}$$

Tāpiri 2030625 ki te 31500.

$$x=\frac{-1425±\sqrt{2062125}}{2\times 7875}$$

Tuhia te pūtakerua o te 2062125.

$$x=\frac{-1425±15\sqrt{9165}}{2\times 7875}$$

Whakareatia 2 ki te 7875.

$$x=\frac{-1425±15\sqrt{9165}}{15750}$$

Nā, me whakaoti te whārite x=\frac{-1425±15\sqrt{9165}}{15750} ina he tāpiri te ±. Tāpiri -1425 ki te 15\sqrt{9165}.

$$x=\frac{15\sqrt{9165}-1425}{15750}$$

Whakawehe -1425+15\sqrt{9165} ki te 15750.

$$x=\frac{\sqrt{9165}}{1050}-\frac{19}{210}$$

Nā, me whakaoti te whārite x=\frac{-1425±15\sqrt{9165}}{15750} ina he tango te ±. Tango 15\sqrt{9165} mai i -1425.

$$x=\frac{-15\sqrt{9165}-1425}{15750}$$

Whakawehe -1425-15\sqrt{9165} ki te 15750.

$$x=-\frac{\sqrt{9165}}{1050}-\frac{19}{210}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{9165}}{1050}-\frac{19}{210} x=-\frac{\sqrt{9165}}{1050}-\frac{19}{210}$$

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.

$$7875x^{2}+1425x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$7875x^{2}+1425x-1-\left(-1\right)=-\left(-1\right)$$

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

$$7875x^{2}+1425x=-\left(-1\right)$$

Tango -1 mai i 0.

$$7875x^{2}+1425x=1$$

Whakawehea ngā taha e rua ki te 7875.

$$\frac{7875x^{2}+1425x}{7875}=\frac{1}{7875}$$

Mā te whakawehe ki te 7875 ka wetekia te whakareanga ki te 7875.

$$x^{2}+\frac{1425}{7875}x=\frac{1}{7875}$$

Whakahekea te hautanga \frac{1425}{7875} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 75.

$$x^{2}+\frac{19}{105}x=\frac{1}{7875}$$

Whakawehea te \frac{19}{105}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{19}{210}. Nā, tāpiria te pūrua o te \frac{19}{210} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{19}{105}x+\left(\frac{19}{210}\right)^{2}=\frac{1}{7875}+\left(\frac{19}{210}\right)^{2}$$

Pūruatia \frac{19}{210} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{19}{105}x+\frac{361}{44100}=\frac{1}{7875}+\frac{361}{44100}$$

Tāpiri \frac{1}{7875} ki te \frac{361}{44100} 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.

$$x^{2}+\frac{19}{105}x+\frac{361}{44100}=\frac{611}{73500}$$

Tauwehea x^{2}+\frac{19}{105}x+\frac{361}{44100}. 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}.

$$\left(x+\frac{19}{210}\right)^{2}=\frac{611}{73500}$$

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

$$\sqrt{\left(x+\frac{19}{210}\right)^{2}}=\sqrt{\frac{611}{73500}}$$

Whakarūnātia.

$$x+\frac{19}{210}=\frac{\sqrt{9165}}{1050} x+\frac{19}{210}=-\frac{\sqrt{9165}}{1050}$$

Me tango \frac{19}{210} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{9165}}{1050}-\frac{19}{210} x=-\frac{\sqrt{9165}}{1050}-\frac{19}{210}$$