Whakaoti i te 4/x^2-55/x=14

$$\frac { 4 } { x ^ { 2 } } - \frac { 55 } { x } = 14$$

Whakaoti mō x

$$x=-4$$

$$x=\frac{1}{14}\approx 0.071428571$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$4-x\times 55=14x^{2}$$

Tangohia te 14x^{2} mai i ngā taha e rua.

$$4-x\times 55-14x^{2}=0$$

Whakareatia te -1 ki te 55, ka -55.

$$4-55x-14x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-14x^{2}-55x+4=0$$

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.

$$a+b=-55 ab=-14\times 4=-56$$

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 -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 -55.

$$a=1 b=-56$$

Tuhia anō te -14x^{2}-55x+4 hei \left(-14x^{2}+x\right)+\left(-56x+4\right).

$$\left(-14x^{2}+x\right)+\left(-56x+4\right)$$

Tauwehea te -x i te tuatahi me te -4 i te rōpū tuarua.

$$-x\left(14x-1\right)-4\left(14x-1\right)$$

Whakatauwehea atu te kīanga pātahi 14x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(14x-1\right)\left(-x-4\right)$$

Hei kimi otinga whārite, me whakaoti te 14x-1=0 me te -x-4=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$4-x\times 55=14x^{2}$$

Tangohia te 14x^{2} mai i ngā taha e rua.

$$4-x\times 55-14x^{2}=0$$

Whakareatia te -1 ki te 55, ka -55.

$$4-55x-14x^{2}=0$$

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.

$$-14x^{2}-55x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -14 mō a, -55 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Pūrua -55.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\left(-14\right)\times 4}}{2\left(-14\right)}$$

Whakareatia -4 ki te -14.

$$x=\frac{-\left(-55\right)±\sqrt{3025+56\times 4}}{2\left(-14\right)}$$

Whakareatia 56 ki te 4.

$$x=\frac{-\left(-55\right)±\sqrt{3025+224}}{2\left(-14\right)}$$

Tāpiri 3025 ki te 224.

$$x=\frac{-\left(-55\right)±\sqrt{3249}}{2\left(-14\right)}$$

Tuhia te pūtakerua o te 3249.

$$x=\frac{-\left(-55\right)±57}{2\left(-14\right)}$$

Ko te tauaro o -55 ko 55.

$$x=\frac{55±57}{2\left(-14\right)}$$

Whakareatia 2 ki te -14.

$$x=\frac{55±57}{-28}$$

Nā, me whakaoti te whārite x=\frac{55±57}{-28} ina he tāpiri te ±. Tāpiri 55 ki te 57.

$$x=\frac{112}{-28}$$

Whakawehe 112 ki te -28.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{55±57}{-28} ina he tango te ±. Tango 57 mai i 55.

$$x=-\frac{2}{-28}$$

Whakahekea te hautanga \frac{-2}{-28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{14}$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2},x.

$$4-x\times 55=14x^{2}$$

Tangohia te 14x^{2} mai i ngā taha e rua.

$$4-x\times 55-14x^{2}=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x\times 55-14x^{2}=-4$$

Whakareatia te -1 ki te 55, ka -55.

$$-55x-14x^{2}=-4$$

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.

$$-14x^{2}-55x=-4$$

Whakawehea ngā taha e rua ki te -14.

$$\frac{-14x^{2}-55x}{-14}=-\frac{4}{-14}$$

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

$$x^{2}+\left(-\frac{55}{-14}\right)x=-\frac{4}{-14}$$

Whakawehe -55 ki te -14.

$$x^{2}+\frac{55}{14}x=-\frac{4}{-14}$$

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

$$x^{2}+\frac{55}{14}x=\frac{2}{7}$$

Whakawehea te \frac{55}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{55}{28}. Nā, tāpiria te pūrua o te \frac{55}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{55}{14}x+\left(\frac{55}{28}\right)^{2}=\frac{2}{7}+\left(\frac{55}{28}\right)^{2}$$

Pūruatia \frac{55}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{55}{14}x+\frac{3025}{784}=\frac{2}{7}+\frac{3025}{784}$$

Tāpiri \frac{2}{7} ki te \frac{3025}{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.

$$x^{2}+\frac{55}{14}x+\frac{3025}{784}=\frac{3249}{784}$$

Tauwehea x^{2}+\frac{55}{14}x+\frac{3025}{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}.

$$\left(x+\frac{55}{28}\right)^{2}=\frac{3249}{784}$$

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

$$\sqrt{\left(x+\frac{55}{28}\right)^{2}}=\sqrt{\frac{3249}{784}}$$

Whakarūnātia.

$$x+\frac{55}{28}=\frac{57}{28} x+\frac{55}{28}=-\frac{57}{28}$$

Me tango \frac{55}{28} mai i ngā taha e rua o te whārite.

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