Whakaoti i te 7.5/x=7.5/x+15+1/4

$$\frac { 7.5 } { x } = \frac { 7.5 } { x + 15 } + \frac { 1 } { 4 }$$

Whakaoti mō x

$$x=-30$$

$$x=15$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -15,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x\left(x+15\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x+60 ki te 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te 7.5, ka 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te \frac{1}{4}, ka 1.

$$30x+450=30x+x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+15.

$$30x+450=30x+x^{2}+15x$$

Pahekotia te 30x me 15x, ka 45x.

$$30x+450=45x+x^{2}$$

Tangohia te 45x mai i ngā taha e rua.

$$30x+450-45x=x^{2}$$

Pahekotia te 30x me -45x, ka -15x.

$$-15x+450=x^{2}$$

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

$$-15x+450-x^{2}=0$$

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

$$-x^{2}-15x+450=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+450. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-15 ab=-450=-450$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -450.

$$1,-450 2,-225 3,-150 5,-90 6,-75 9,-50 10,-45 15,-30 18,-25$$

Tātaihia te tapeke mō ia takirua.

$$1-450=-449 2-225=-223 3-150=-147 5-90=-85 6-75=-69 9-50=-41 10-45=-35 15-30=-15 18-25=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -15.

$$a=15 b=-30$$

Tuhia anō te -x^{2}-15x+450 hei \left(-x^{2}+15x\right)+\left(-30x+450\right).

$$\left(-x^{2}+15x\right)+\left(-30x+450\right)$$

Tauwehea te x i te tuatahi me te 30 i te rōpū tuarua.

$$x\left(-x+15\right)+30\left(-x+15\right)$$

Whakatauwehea atu te kīanga pātahi -x+15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+15\right)\left(x+30\right)$$

Hei kimi otinga whārite, me whakaoti te -x+15=0 me te x+30=0.

$$x=15 x=-30$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -15,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x\left(x+15\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x+60 ki te 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te 7.5, ka 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te \frac{1}{4}, ka 1.

$$30x+450=30x+x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+15.

$$30x+450=30x+x^{2}+15x$$

Pahekotia te 30x me 15x, ka 45x.

$$30x+450=45x+x^{2}$$

Tangohia te 45x mai i ngā taha e rua.

$$30x+450-45x=x^{2}$$

Pahekotia te 30x me -45x, ka -15x.

$$-15x+450=x^{2}$$

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

$$-15x+450-x^{2}=0$$

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.

$$-x^{2}-15x+450=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -15 mō b, me 450 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-1\right)\times 450}}{2\left(-1\right)}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-1\right)\times 450}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-15\right)±\sqrt{225+4\times 450}}{2\left(-1\right)}$$

Whakareatia 4 ki te 450.

$$x=\frac{-\left(-15\right)±\sqrt{225+1800}}{2\left(-1\right)}$$

Tāpiri 225 ki te 1800.

$$x=\frac{-\left(-15\right)±\sqrt{2025}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 2025.

$$x=\frac{-\left(-15\right)±45}{2\left(-1\right)}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±45}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{15±45}{-2}$$

Nā, me whakaoti te whārite x=\frac{15±45}{-2} ina he tāpiri te ±. Tāpiri 15 ki te 45.

$$x=\frac{60}{-2}$$

Whakawehe 60 ki te -2.

$$x=-30$$

Nā, me whakaoti te whārite x=\frac{15±45}{-2} ina he tango te ±. Tango 45 mai i 15.

$$x=-\frac{30}{-2}$$

Whakawehe -30 ki te -2.

$$x=15$$

Kua oti te whārite te whakatau.

$$x=-30 x=15$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -15,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4x\left(x+15\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+15,4.

$$\left(4x+60\right)\times 7.5=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4x+60 ki te 7.5.

$$30x+450=4x\times 7.5+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te 7.5, ka 30.

$$30x+450=30x+4x\left(x+15\right)\times \frac{1}{4}$$

Whakareatia te 4 ki te \frac{1}{4}, ka 1.

$$30x+450=30x+x\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+15.

$$30x+450=30x+x^{2}+15x$$

Pahekotia te 30x me 15x, ka 45x.

$$30x+450=45x+x^{2}$$

Tangohia te 45x mai i ngā taha e rua.

$$30x+450-45x=x^{2}$$

Pahekotia te 30x me -45x, ka -15x.

$$-15x+450=x^{2}$$

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

$$-15x+450-x^{2}=0$$

Tangohia te 450 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-15x-x^{2}=-450$$

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.

$$-x^{2}-15x=-450$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-15x}{-1}=-\frac{450}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{15}{-1}\right)x=-\frac{450}{-1}$$

Whakawehe -15 ki te -1.

$$x^{2}+15x=-\frac{450}{-1}$$

Whakawehe -450 ki te -1.

$$x^{2}+15x=450$$

Whakawehea te 15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{2}. Nā, tāpiria te pūrua o te \frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=450+\left(\frac{15}{2}\right)^{2}$$

Pūruatia \frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+15x+\frac{225}{4}=450+\frac{225}{4}$$

Tāpiri 450 ki te \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{2025}{4}$$

Tauwehea x^{2}+15x+\frac{225}{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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{2025}{4}$$

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

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{2025}{4}}$$

Whakarūnātia.

$$x+\frac{15}{2}=\frac{45}{2} x+\frac{15}{2}=-\frac{45}{2}$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$x=15 x=-30$$