Whakaoti i te 16/x^2-9+4/x^2-5x+6-5(x+2)/6-x-x^2=0

$$\frac { 16 } { x ^ { 2 } - 9 } + \frac { 4 } { x ^ { 2 } - 5 x + 6 } - \frac { 5 ( x + 2 ) } { 6 - x - x ^ { 2 } } = 0$$

Whakaoti mō x

$$x=-5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,2,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x-2\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 4.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Pahekotia te 16x me 4x, ka 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Tāpirihia te -32 ki te 12, ka -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-5x ki te x+2 ka whakakotahi i ngā kupu rite.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

Hei kimi i te tauaro o 5x+30-5x^{2}, kimihia te tauaro o ia taurangi.

$$20x-20-5x-30+5x^{2}=0$$

Pahekotia te 20x me -5x, ka 15x.

$$15x-20-30+5x^{2}=0$$

Tangohia te 30 i te -20, ka -50.

$$15x-50+5x^{2}=0$$

Whakawehea ngā taha e rua ki te 5.

$$3x-10+x^{2}=0$$

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

$$x^{2}+3x-10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=1\left(-10\right)=-10$$

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 -10.

$$-1,10 -2,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+10=9 -2+5=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-2 b=5$$

Tuhia anō te x^{2}+3x-10 hei \left(x^{2}-2x\right)+\left(5x-10\right).

$$\left(x^{2}-2x\right)+\left(5x-10\right)$$

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

$$x\left(x-2\right)+5\left(x-2\right)$$

Whakatauwehea atu te kīanga pātahi x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-2\right)\left(x+5\right)$$

Hei kimi otinga whārite, me whakaoti te x-2=0 me te x+5=0.

$$x=2 x=-5$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,2,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x-2\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 4.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Pahekotia te 16x me 4x, ka 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Tāpirihia te -32 ki te 12, ka -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-5x ki te x+2 ka whakakotahi i ngā kupu rite.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

Hei kimi i te tauaro o 5x+30-5x^{2}, kimihia te tauaro o ia taurangi.

$$20x-20-5x-30+5x^{2}=0$$

Pahekotia te 20x me -5x, ka 15x.

$$15x-20-30+5x^{2}=0$$

Tangohia te 30 i te -20, ka -50.

$$15x-50+5x^{2}=0$$

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.

$$5x^{2}+15x-50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 15 mō b, me -50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5\left(-50\right)}}{2\times 5}$$

Pūrua 15.

$$x=\frac{-15±\sqrt{225-4\times 5\left(-50\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-15±\sqrt{225-20\left(-50\right)}}{2\times 5}$$

Whakareatia -20 ki te -50.

$$x=\frac{-15±\sqrt{225+1000}}{2\times 5}$$

Tāpiri 225 ki te 1000.

$$x=\frac{-15±\sqrt{1225}}{2\times 5}$$

Tuhia te pūtakerua o te 1225.

$$x=\frac{-15±35}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{-15±35}{10}$$

Nā, me whakaoti te whārite x=\frac{-15±35}{10} ina he tāpiri te ±. Tāpiri -15 ki te 35.

$$x=\frac{20}{10}$$

Whakawehe 20 ki te 10.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-15±35}{10} ina he tango te ±. Tango 35 mai i -15.

$$x=-\frac{50}{10}$$

Whakawehe -50 ki te 10.

$$x=-5$$

Kua oti te whārite te whakatau.

$$x=2 x=-5$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,2,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x-2\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 4.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Pahekotia te 16x me 4x, ka 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Tāpirihia te -32 ki te 12, ka -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 15-5x ki te x+2 ka whakakotahi i ngā kupu rite.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

Hei kimi i te tauaro o 5x+30-5x^{2}, kimihia te tauaro o ia taurangi.

$$20x-20-5x-30+5x^{2}=0$$

Pahekotia te 20x me -5x, ka 15x.

$$15x-20-30+5x^{2}=0$$

Tangohia te 30 i te -20, ka -50.

$$15x-50+5x^{2}=0$$

Me tāpiri te 50 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$15x+5x^{2}=50$$

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.

$$5x^{2}+15x=50$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}+15x}{5}=\frac{50}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\frac{15}{5}x=\frac{50}{5}$$

Whakawehe 15 ki te 5.

$$x^{2}+3x=\frac{50}{5}$$

Whakawehe 50 ki te 5.

$$x^{2}+3x=10$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

Tauwehea x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x+\frac{3}{2}=\frac{7}{2} x+\frac{3}{2}=-\frac{7}{2}$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=2 x=-5$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-5$$