Whakaoti i te 1+6/x=-5/x^2

$$1 + \frac { 6 } { x } = - \frac { 5 } { x ^ { 2 } }$$

Whakaoti mō x

$$x=-5$$

$$x=-1$$

Ngā Upane Whakamahi Tauwehe

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,x^{2}.

$$x^{2}+x\times 6=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}+x\times 6+5=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+6x+5 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=5$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=5$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

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

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

$$x=-1 x=-5$$

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,x^{2}.

$$x^{2}+x\times 6=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}+x\times 6+5=0$$

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

$$a+b=6 ab=1\times 5=5$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=5$$

Tuhia anō te x^{2}+6x+5 hei \left(x^{2}+x\right)+\left(5x+5\right).

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

Tauwehea te x i te tuatahi me te 5 i te rōpū tuarua.

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

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

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

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

$$x=-1 x=-5$$

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,x^{2}.

$$x^{2}+x\times 6=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$x^{2}+x\times 6+5=0$$

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.

$$x^{2}+6x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 5}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 5}}{2}$$

Whakareatia -4 ki te 5.

$$x=\frac{-6±\sqrt{36-20}}{2}$$

Tāpiri 36 ki te -20.

$$x=\frac{-6±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-6±4}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±4}{2} ina he tāpiri te ±. Tāpiri -6 ki te 4.

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

Whakawehe -2 ki te 2.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-6±4}{2} ina he tango te ±. Tango 4 mai i -6.

$$x=-\frac{10}{2}$$

Whakawehe -10 ki te 2.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=-1 x=-5$$

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,x^{2}.

$$x^{2}+x\times 6=-5$$

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.

$$x^{2}+6x=-5$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+6x+3^{2}=-5+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-5+9$$

Tāpiri -5 ki te 9.

$$x^{2}+6x+9=4$$

Tauwehea x^{2}+6x+9. 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}.

$$\left(x+3\right)^{2}=4$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{4}$$

Whakarūnātia.

$$x+3=2 x+3=-2$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x=-1 x=-5$$