Whakaoti i te 3/x+3/x-5=3x-12/x-5

$$\frac { 3 } { x } + \frac { 3 } { x - 5 } = \frac { 3 x - 12 } { x - 5 }$$

Whakaoti mō x

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-5.

$$\left(x-5\right)\times 3+x\times 3=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 3.

$$3x-15+x\times 3=x\left(3x-12\right)$$

Pahekotia te 3x me x\times 3, ka 6x.

$$6x-15=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 3x-12.

$$6x-15=3x^{2}-12x$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6x-15-3x^{2}=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$6x-15-3x^{2}+12x=0$$

Pahekotia te 6x me 12x, ka 18x.

$$18x-15-3x^{2}=0$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

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.

$$a+b=6 ab=-\left(-5\right)=5$$

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.

$$a=5 b=1$$

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

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

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

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

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

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

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

$$x=5 x=1$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-5.

$$\left(x-5\right)\times 3+x\times 3=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 3.

$$3x-15+x\times 3=x\left(3x-12\right)$$

Pahekotia te 3x me x\times 3, ka 6x.

$$6x-15=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 3x-12.

$$6x-15=3x^{2}-12x$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6x-15-3x^{2}=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$6x-15-3x^{2}+12x=0$$

Pahekotia te 6x me 12x, ka 18x.

$$18x-15-3x^{2}=0$$

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.

$$-3x^{2}+18x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 18 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-3\right)\left(-15\right)}}{2\left(-3\right)}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\left(-3\right)\left(-15\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-18±\sqrt{324+12\left(-15\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -15.

$$x=\frac{-18±\sqrt{324-180}}{2\left(-3\right)}$$

Tāpiri 324 ki te -180.

$$x=\frac{-18±\sqrt{144}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-18±12}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-18±12}{-6}$$

Nā, me whakaoti te whārite x=\frac{-18±12}{-6} ina he tāpiri te ±. Tāpiri -18 ki te 12.

$$x=-\frac{6}{-6}$$

Whakawehe -6 ki te -6.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{-18±12}{-6} ina he tango te ±. Tango 12 mai i -18.

$$x=-\frac{30}{-6}$$

Whakawehe -30 ki te -6.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=1 x=5$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-5.

$$\left(x-5\right)\times 3+x\times 3=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 3.

$$3x-15+x\times 3=x\left(3x-12\right)$$

Pahekotia te 3x me x\times 3, ka 6x.

$$6x-15=x\left(3x-12\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te 3x-12.

$$6x-15=3x^{2}-12x$$

Tangohia te 3x^{2} mai i ngā taha e rua.

$$6x-15-3x^{2}=-12x$$

Me tāpiri te 12x ki ngā taha e rua.

$$6x-15-3x^{2}+12x=0$$

Pahekotia te 6x me 12x, ka 18x.

$$18x-15-3x^{2}=0$$

Me tāpiri te 15 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$18x-3x^{2}=15$$

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.

$$-3x^{2}+18x=15$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+18x}{-3}=\frac{15}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{18}{-3}x=\frac{15}{-3}$$

Whakawehe 18 ki te -3.

$$x^{2}-6x=\frac{15}{-3}$$

Whakawehe 15 ki te -3.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=-5+\left(-3\right)^{2}$$

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 tāpiri 3 ki ngā taha e rua o te whārite.

$$x=5 x=1$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=1$$