Whakaoti i te x+9/x+16x/x+9=8

$$\frac { x + 9 } { x } + \frac { 16 x } { x + 9 } = 8$$

Whakaoti mō x

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -9,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+9\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+9.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x+9 ki te x+9, ka \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Pahekotia te x^{2} me x^{2}\times 16, ka 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$17x^{2}+18x+81-8x^{2}=72x$$

Pahekotia te 17x^{2} me -8x^{2}, ka 9x^{2}.

$$9x^{2}+18x+81=72x$$

Tangohia te 72x mai i ngā taha e rua.

$$9x^{2}+18x+81-72x=0$$

Pahekotia te 18x me -72x, ka -54x.

$$9x^{2}-54x+81=0$$

Whakawehea ngā taha e rua ki te 9.

$$x^{2}-6x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 9=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-3 b=-3$$

Tuhia anō te x^{2}-6x+9 hei \left(x^{2}-3x\right)+\left(-3x+9\right).

$$\left(x^{2}-3x\right)+\left(-3x+9\right)$$

Tauwehea te x i te tuatahi me te -3 i te rōpū tuarua.

$$x\left(x-3\right)-3\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(x-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x-3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x-3=0.

$$x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -9,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+9\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+9.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x+9 ki te x+9, ka \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Pahekotia te x^{2} me x^{2}\times 16, ka 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$17x^{2}+18x+81-8x^{2}=72x$$

Pahekotia te 17x^{2} me -8x^{2}, ka 9x^{2}.

$$9x^{2}+18x+81=72x$$

Tangohia te 72x mai i ngā taha e rua.

$$9x^{2}+18x+81-72x=0$$

Pahekotia te 18x me -72x, ka -54x.

$$9x^{2}-54x+81=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -54 mō b, me 81 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 9\times 81}}{2\times 9}$$

Pūrua -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 9\times 81}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-54\right)±\sqrt{2916-36\times 81}}{2\times 9}$$

Whakareatia -36 ki te 81.

$$x=\frac{-\left(-54\right)±\sqrt{2916-2916}}{2\times 9}$$

Tāpiri 2916 ki te -2916.

$$x=\frac{-\left(-54\right)±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-54}{2\times 9}$$

Ko te tauaro o -54 ko 54.

$$x=\frac{54}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{54}{18}$$

Whakawehe 54 ki te 18.

$$x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -9,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+9\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+9.

$$\left(x+9\right)\left(x+9\right)+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x+9 ki te x+9, ka \left(x+9\right)^{2}.

$$\left(x+9\right)^{2}+x\times 16x=8x\left(x+9\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+9\right)^{2}.

$$x^{2}+18x+81+x\times 16x=8x\left(x+9\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}+18x+81+x^{2}\times 16=8x\left(x+9\right)$$

Pahekotia te x^{2} me x^{2}\times 16, ka 17x^{2}.

$$17x^{2}+18x+81=8x\left(x+9\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x+9.

$$17x^{2}+18x+81=8x^{2}+72x$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$17x^{2}+18x+81-8x^{2}=72x$$

Pahekotia te 17x^{2} me -8x^{2}, ka 9x^{2}.

$$9x^{2}+18x+81=72x$$

Tangohia te 72x mai i ngā taha e rua.

$$9x^{2}+18x+81-72x=0$$

Pahekotia te 18x me -72x, ka -54x.

$$9x^{2}-54x+81=0$$

Tangohia te 81 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9x^{2}-54x=-81$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-54x}{9}=-\frac{81}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\left(-\frac{54}{9}\right)x=-\frac{81}{9}$$

Whakawehe -54 ki te 9.

$$x^{2}-6x=-\frac{81}{9}$$

Whakawehe -81 ki te 9.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Pūrua -3.

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

Tāpiri -9 ki te 9.

$$x^{2}-6x+9=0$$

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}.

$$\left(x-3\right)^{2}=0$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-3=0 x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=3 x=3$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=3$$