Whakaoti i te 4x^2-14x-8=0

$$4 x ^ { 2 } - 14 x - 8 = 0$$

Whakaoti mō x

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

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$2x^{2}-7x-4=0$$

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

$$a+b=-7 ab=2\left(-4\right)=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-8 b=1$$

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

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

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

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

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

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

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

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

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.

$$4x^{2}-14x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -14 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\left(-8\right)}}{2\times 4}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 4\left(-8\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-14\right)±\sqrt{196-16\left(-8\right)}}{2\times 4}$$

Whakareatia -16 ki te -8.

$$x=\frac{-\left(-14\right)±\sqrt{196+128}}{2\times 4}$$

Tāpiri 196 ki te 128.

$$x=\frac{-\left(-14\right)±\sqrt{324}}{2\times 4}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-\left(-14\right)±18}{2\times 4}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±18}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{14±18}{8}$$

Nā, me whakaoti te whārite x=\frac{14±18}{8} ina he tāpiri te ±. Tāpiri 14 ki te 18.

$$x=\frac{32}{8}$$

Whakawehe 32 ki te 8.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{14±18}{8} ina he tango te ±. Tango 18 mai i 14.

$$x=-\frac{4}{8}$$

Whakahekea te hautanga \frac{-4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

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

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.

$$4x^{2}-14x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

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

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

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

Tango -8 mai i 0.

$$4x^{2}-14x=8$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-14x}{4}=\frac{8}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

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

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

$$x^{2}-\frac{7}{2}x=\frac{8}{4}$$

Whakawehe 8 ki te 4.

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

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=2+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=2+\frac{49}{16}$$

Tāpiri 2 ki te \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{81}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{9}{4} x-\frac{7}{4}=-\frac{9}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

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