Whakaoti i te x/x-5+6/x+7=12x/(x-5)(x+7)

$$\frac { x } { x - 5 } + \frac { 6 } { x + 7 } = \frac { 12 x } { ( x - 5 ) ( x + 7 ) }$$

Whakaoti mō x

$$x=-6$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -7,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+7,\left(x-5\right)\left(x+7\right).

$$\left(x+7\right)x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te x.

$$x^{2}+7x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 6.

$$x^{2}+7x+6x-30=12x$$

Pahekotia te 7x me 6x, ka 13x.

$$x^{2}+13x-30=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+13x-30-12x=0$$

Pahekotia te 13x me -12x, ka x.

$$x^{2}+x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+x-30 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.

$$a+b=1 ab=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

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

$$a=-5 b=6$$

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

$$\left(x-5\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+6=0.

$$x=5 x=-6$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=-6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -7,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+7,\left(x-5\right)\left(x+7\right).

$$\left(x+7\right)x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te x.

$$x^{2}+7x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 6.

$$x^{2}+7x+6x-30=12x$$

Pahekotia te 7x me 6x, ka 13x.

$$x^{2}+13x-30=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+13x-30-12x=0$$

Pahekotia te 13x me -12x, ka x.

$$x^{2}+x-30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-30\right)=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

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

$$a=-5 b=6$$

Tuhia anō te x^{2}+x-30 hei \left(x^{2}-5x\right)+\left(6x-30\right).

$$\left(x^{2}-5x\right)+\left(6x-30\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-5\right)+6\left(x-5\right)$$

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

$$\left(x-5\right)\left(x+6\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+6=0.

$$x=5 x=-6$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -7,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+7,\left(x-5\right)\left(x+7\right).

$$\left(x+7\right)x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te x.

$$x^{2}+7x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 6.

$$x^{2}+7x+6x-30=12x$$

Pahekotia te 7x me 6x, ka 13x.

$$x^{2}+13x-30=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+13x-30-12x=0$$

Pahekotia te 13x me -12x, ka x.

$$x^{2}+x-30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-30\right)}}{2}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-30\right)}}{2}$$

Whakareatia -4 ki te -30.

$$x=\frac{-1±\sqrt{1+120}}{2}$$

Tāpiri 1 ki te 120.

$$x=\frac{-1±\sqrt{121}}{2}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-1±11}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±11}{2} ina he tāpiri te ±. Tāpiri -1 ki te 11.

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

Whakawehe 10 ki te 2.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{-1±11}{2} ina he tango te ±. Tango 11 mai i -1.

$$x=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$x=-6$$

Kua oti te whārite te whakatau.

$$x=5 x=-6$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -7,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-5\right)\left(x+7\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-5,x+7,\left(x-5\right)\left(x+7\right).

$$\left(x+7\right)x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te x.

$$x^{2}+7x+\left(x-5\right)\times 6=12x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-5 ki te 6.

$$x^{2}+7x+6x-30=12x$$

Pahekotia te 7x me 6x, ka 13x.

$$x^{2}+13x-30=12x$$

Tangohia te 12x mai i ngā taha e rua.

$$x^{2}+13x-30-12x=0$$

Pahekotia te 13x me -12x, ka x.

$$x^{2}+x-30=0$$

Me tāpiri te 30 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}+x=30$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=30+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=30+\frac{1}{4}$$

Tāpiri 30 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{121}{4}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{11}{2} x+\frac{1}{2}=-\frac{11}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=5 x=-6$$

Tē taea kia ōrite te tāupe x ki 5.

$$x=-6$$