Whakaoti i te 4x^2-x=5

$$4 x ^ { 2 } - x = 5$$

Whakaoti mō x

$$x=-1$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 5 mai i ngā taha e rua.

$$4x^{2}-x-5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=4\left(-5\right)=-20$$

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 -20.

$$1,-20 2,-10 4,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-20=-19 2-10=-8 4-5=-1$$

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

$$a=-5 b=4$$

Tuhia anō te 4x^{2}-x-5 hei \left(4x^{2}-5x\right)+\left(4x-5\right).

$$\left(4x^{2}-5x\right)+\left(4x-5\right)$$

Whakatauwehea atu x i te 4x^{2}-5x.

$$x\left(4x-5\right)+4x-5$$

Whakatauwehea atu te kīanga pātahi 4x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-5\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 4x-5=0 me te x+1=0.

$$x=\frac{5}{4} x=-1$$

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.

$$4x^{2}-x=5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$4x^{2}-x-5=5-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$4x^{2}-x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -1 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 4\left(-5\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-1\right)±\sqrt{1-16\left(-5\right)}}{2\times 4}$$

Whakareatia -16 ki te -5.

$$x=\frac{-\left(-1\right)±\sqrt{1+80}}{2\times 4}$$

Tāpiri 1 ki te 80.

$$x=\frac{-\left(-1\right)±\sqrt{81}}{2\times 4}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-\left(-1\right)±9}{2\times 4}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±9}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{1±9}{8}$$

Nā, me whakaoti te whārite x=\frac{1±9}{8} ina he tāpiri te ±. Tāpiri 1 ki te 9.

$$x=\frac{10}{8}$$

Whakahekea te hautanga \frac{10}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{5}{4}$$

Nā, me whakaoti te whārite x=\frac{1±9}{8} ina he tango te ±. Tango 9 mai i 1.

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

Whakawehe -8 ki te 8.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{4} x=-1$$

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.

$$4x^{2}-x=5$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-x}{4}=\frac{5}{4}$$

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

$$x^{2}-\frac{1}{4}x=\frac{5}{4}$$

Whakawehea te -\frac{1}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{8}. Nā, tāpiria te pūrua o te -\frac{1}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\frac{5}{4}+\left(-\frac{1}{8}\right)^{2}$$

Pūruatia -\frac{1}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{5}{4}+\frac{1}{64}$$

Tāpiri \frac{5}{4} ki te \frac{1}{64} 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.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{81}{64}$$

Tauwehea x^{2}-\frac{1}{4}x+\frac{1}{64}. 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}.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{81}{64}$$

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

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Whakarūnātia.

$$x-\frac{1}{8}=\frac{9}{8} x-\frac{1}{8}=-\frac{9}{8}$$

Me tāpiri \frac{1}{8} ki ngā taha e rua o te whārite.

$$x=\frac{5}{4} x=-1$$