Whakaoti i te -14x^2+x+4=0

$$- 14 x ^ { 2 } + x + 4 = 0$$

Whakaoti mō x

$$x=\frac{4}{7}\approx 0.571428571$$

$$x=-\frac{1}{2}=-0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -14x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-14\times 4=-56$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -56.

$$-1,56 -2,28 -4,14 -7,8$$

Tātaihia te tapeke mō ia takirua.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=8 b=-7$$

Tuhia anō te -14x^{2}+x+4 hei \left(-14x^{2}+8x\right)+\left(-7x+4\right).

$$\left(-14x^{2}+8x\right)+\left(-7x+4\right)$$

Whakatauwehea atu 2x i te -14x^{2}+8x.

$$2x\left(-7x+4\right)-7x+4$$

Whakatauwehea atu te kīanga pātahi -7x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-7x+4\right)\left(2x+1\right)$$

Hei kimi otinga whārite, me whakaoti te -7x+4=0 me te 2x+1=0.

$$x=\frac{4}{7} x=-\frac{1}{2}$$

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.

$$-14x^{2}+x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -14 mō a, 1 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Whakareatia -4 ki te -14.

$$x=\frac{-1±\sqrt{1+56\times 4}}{2\left(-14\right)}$$

Whakareatia 56 ki te 4.

$$x=\frac{-1±\sqrt{1+224}}{2\left(-14\right)}$$

Tāpiri 1 ki te 224.

$$x=\frac{-1±\sqrt{225}}{2\left(-14\right)}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-1±15}{2\left(-14\right)}$$

Whakareatia 2 ki te -14.

$$x=\frac{-1±15}{-28}$$

Nā, me whakaoti te whārite x=\frac{-1±15}{-28} ina he tāpiri te ±. Tāpiri -1 ki te 15.

$$x=\frac{14}{-28}$$

Whakahekea te hautanga \frac{14}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x=-\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±15}{-28} ina he tango te ±. Tango 15 mai i -1.

$$x=-\frac{16}{-28}$$

Whakahekea te hautanga \frac{-16}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{4}{7}$$

Kua oti te whārite te whakatau.

$$x=-\frac{1}{2} x=\frac{4}{7}$$

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.

$$-14x^{2}+x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-14x^{2}+x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$-14x^{2}+x=-4$$

Whakawehea ngā taha e rua ki te -14.

$$\frac{-14x^{2}+x}{-14}=-\frac{4}{-14}$$

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

$$x^{2}+\frac{1}{-14}x=-\frac{4}{-14}$$

Whakawehe 1 ki te -14.

$$x^{2}-\frac{1}{14}x=-\frac{4}{-14}$$

Whakahekea te hautanga \frac{-4}{-14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{14}x=\frac{2}{7}$$

Whakawehea te -\frac{1}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{28}. Nā, tāpiria te pūrua o te -\frac{1}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{14}x+\left(-\frac{1}{28}\right)^{2}=\frac{2}{7}+\left(-\frac{1}{28}\right)^{2}$$

Pūruatia -\frac{1}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{14}x+\frac{1}{784}=\frac{2}{7}+\frac{1}{784}$$

Tāpiri \frac{2}{7} ki te \frac{1}{784} 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.

$$x^{2}-\frac{1}{14}x+\frac{1}{784}=\frac{225}{784}$$

Tauwehea x^{2}-\frac{1}{14}x+\frac{1}{784}. 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}.

$$\left(x-\frac{1}{28}\right)^{2}=\frac{225}{784}$$

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

$$\sqrt{\left(x-\frac{1}{28}\right)^{2}}=\sqrt{\frac{225}{784}}$$

Whakarūnātia.

$$x-\frac{1}{28}=\frac{15}{28} x-\frac{1}{28}=-\frac{15}{28}$$

Me tāpiri \frac{1}{28} ki ngā taha e rua o te whārite.

$$x=\frac{4}{7} x=-\frac{1}{2}$$