Whakaoti i te 3/2+2625+3/2/frac{x+25{2}}=300/x+frac{1}{2}

$$\frac{ 3 }{ 2 } + \frac{ 2625+ \frac{ 3 }{ 2 } }{ \frac{ x+25 }{ 2 } } = \frac{ 300 }{ x } + \frac{ 1 }{ 2 }$$

Whakaoti mō x (complex solution)

$$x=\sqrt{6202621}-2489\approx 1.506173451$$

$$x=-\left(\sqrt{6202621}+2489\right)\approx -4979.506173451$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x, arā, te tauraro pātahi he tino iti rawa te kitea o 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{3}{2}, ka 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Tāpirihia te 2625 ki te \frac{3}{2}, ka \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 4 ki te \frac{5253}{2}, ka 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te 300, ka 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{1}{2}, ka 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Tangohia te 600 mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-600=x$$

Tangohia te x mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-600-x=0$$

Pahekotia te 3x me -x, ka 2x.

$$2x+10506x\left(x+25\right)^{-1}-600=0$$

Whakaraupapatia anō ngā kīanga tau.

$$2x+10506\times \frac{1}{x+25}x-600=0$$

Tē taea kia ōrite te tāupe x ki -25 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+25.

$$2x\left(x+25\right)+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+25.

$$2x^{2}+50x+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Whakareatia te 10506 ki te 1, ka 10506.

$$2x^{2}+50x+10506x+\left(x+25\right)\left(-600\right)=0$$

Pahekotia te 50x me 10506x, ka 10556x.

$$2x^{2}+10556x+\left(x+25\right)\left(-600\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+25 ki te -600.

$$2x^{2}+10556x-600x-15000=0$$

Pahekotia te 10556x me -600x, ka 9956x.

$$2x^{2}+9956x-15000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 9956 mō b, me -15000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9956±\sqrt{9956^{2}-4\times 2\left(-15000\right)}}{2\times 2}$$

Pūrua 9956.

$$x=\frac{-9956±\sqrt{99121936-4\times 2\left(-15000\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-9956±\sqrt{99121936-8\left(-15000\right)}}{2\times 2}$$

Whakareatia -8 ki te -15000.

$$x=\frac{-9956±\sqrt{99121936+120000}}{2\times 2}$$

Tāpiri 99121936 ki te 120000.

$$x=\frac{-9956±\sqrt{99241936}}{2\times 2}$$

Tuhia te pūtakerua o te 99241936.

$$x=\frac{-9956±4\sqrt{6202621}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-9956±4\sqrt{6202621}}{4}$$

Nā, me whakaoti te whārite x=\frac{-9956±4\sqrt{6202621}}{4} ina he tāpiri te ±. Tāpiri -9956 ki te 4\sqrt{6202621}.

$$x=\frac{4\sqrt{6202621}-9956}{4}$$

Whakawehe -9956+4\sqrt{6202621} ki te 4.

$$x=\sqrt{6202621}-2489$$

Nā, me whakaoti te whārite x=\frac{-9956±4\sqrt{6202621}}{4} ina he tango te ±. Tango 4\sqrt{6202621} mai i -9956.

$$x=\frac{-4\sqrt{6202621}-9956}{4}$$

Whakawehe -9956-4\sqrt{6202621} ki te 4.

$$x=-\sqrt{6202621}-2489$$

Kua oti te whārite te whakatau.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x, arā, te tauraro pātahi he tino iti rawa te kitea o 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{3}{2}, ka 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Tāpirihia te 2625 ki te \frac{3}{2}, ka \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 4 ki te \frac{5253}{2}, ka 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te 300, ka 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{1}{2}, ka 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Tangohia te x mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-x=600$$

Pahekotia te 3x me -x, ka 2x.

$$2x+10506x\left(x+25\right)^{-1}=600$$

Whakaraupapatia anō ngā kīanga tau.

$$2x+10506\times \frac{1}{x+25}x=600$$

Tē taea kia ōrite te tāupe x ki -25 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+25.

$$2x\left(x+25\right)+10506\times 1x=600\left(x+25\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+25.

$$2x^{2}+50x+10506\times 1x=600\left(x+25\right)$$

Whakareatia te 10506 ki te 1, ka 10506.

$$2x^{2}+50x+10506x=600\left(x+25\right)$$

Pahekotia te 50x me 10506x, ka 10556x.

$$2x^{2}+10556x=600\left(x+25\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 600 ki te x+25.

$$2x^{2}+10556x=600x+15000$$

Tangohia te 600x mai i ngā taha e rua.

$$2x^{2}+10556x-600x=15000$$

Pahekotia te 10556x me -600x, ka 9956x.

$$2x^{2}+9956x=15000$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+9956x}{2}=\frac{15000}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{9956}{2}x=\frac{15000}{2}$$

Whakawehe 9956 ki te 2.

$$x^{2}+4978x=\frac{15000}{2}$$

Whakawehe 15000 ki te 2.

$$x^{2}+4978x=7500$$

Whakawehea te 4978, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2489. Nā, tāpiria te pūrua o te 2489 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4978x+2489^{2}=7500+2489^{2}$$

Pūrua 2489.

$$x^{2}+4978x+6195121=7500+6195121$$

Tāpiri 7500 ki te 6195121.

$$x^{2}+4978x+6195121=6202621$$

Tauwehea x^{2}+4978x+6195121. 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}.

$$\left(x+2489\right)^{2}=6202621$$

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

$$\sqrt{\left(x+2489\right)^{2}}=\sqrt{6202621}$$

Whakarūnātia.

$$x+2489=\sqrt{6202621} x+2489=-\sqrt{6202621}$$

Me tango 2489 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Whakaoti mō x

$$x=\sqrt{6202621}-2489\approx 1.506173451$$

$$x=-\sqrt{6202621}-2489\approx -4979.506173451$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x, arā, te tauraro pātahi he tino iti rawa te kitea o 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{3}{2}, ka 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Tāpirihia te 2625 ki te \frac{3}{2}, ka \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 4 ki te \frac{5253}{2}, ka 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te 300, ka 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{1}{2}, ka 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Tangohia te 600 mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-600=x$$

Tangohia te x mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-600-x=0$$

Pahekotia te 3x me -x, ka 2x.

$$2x+10506x\left(x+25\right)^{-1}-600=0$$

Whakaraupapatia anō ngā kīanga tau.

$$2x+10506\times \frac{1}{x+25}x-600=0$$

Tē taea kia ōrite te tāupe x ki -25 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+25.

$$2x\left(x+25\right)+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+25.

$$2x^{2}+50x+10506\times 1x+\left(x+25\right)\left(-600\right)=0$$

Whakareatia te 10506 ki te 1, ka 10506.

$$2x^{2}+50x+10506x+\left(x+25\right)\left(-600\right)=0$$

Pahekotia te 50x me 10506x, ka 10556x.

$$2x^{2}+10556x+\left(x+25\right)\left(-600\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+25 ki te -600.

$$2x^{2}+10556x-600x-15000=0$$

Pahekotia te 10556x me -600x, ka 9956x.

$$2x^{2}+9956x-15000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 9956 mō b, me -15000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9956±\sqrt{9956^{2}-4\times 2\left(-15000\right)}}{2\times 2}$$

Pūrua 9956.

$$x=\frac{-9956±\sqrt{99121936-4\times 2\left(-15000\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-9956±\sqrt{99121936-8\left(-15000\right)}}{2\times 2}$$

Whakareatia -8 ki te -15000.

$$x=\frac{-9956±\sqrt{99121936+120000}}{2\times 2}$$

Tāpiri 99121936 ki te 120000.

$$x=\frac{-9956±\sqrt{99241936}}{2\times 2}$$

Tuhia te pūtakerua o te 99241936.

$$x=\frac{-9956±4\sqrt{6202621}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-9956±4\sqrt{6202621}}{4}$$

Nā, me whakaoti te whārite x=\frac{-9956±4\sqrt{6202621}}{4} ina he tāpiri te ±. Tāpiri -9956 ki te 4\sqrt{6202621}.

$$x=\frac{4\sqrt{6202621}-9956}{4}$$

Whakawehe -9956+4\sqrt{6202621} ki te 4.

$$x=\sqrt{6202621}-2489$$

Nā, me whakaoti te whārite x=\frac{-9956±4\sqrt{6202621}}{4} ina he tango te ±. Tango 4\sqrt{6202621} mai i -9956.

$$x=\frac{-4\sqrt{6202621}-9956}{4}$$

Whakawehe -9956-4\sqrt{6202621} ki te 4.

$$x=-\sqrt{6202621}-2489$$

Kua oti te whārite te whakatau.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x, arā, te tauraro pātahi he tino iti rawa te kitea o 2,x.

$$2x\times \frac{3}{2}+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{3}{2}, ka 3.

$$3x+4x\left(x+25\right)^{-1}\left(2625+\frac{3}{2}\right)=2\times 300+2x\times \frac{1}{2}$$

Tāpirihia te 2625 ki te \frac{3}{2}, ka \frac{5253}{2}.

$$3x+4x\left(x+25\right)^{-1}\times \frac{5253}{2}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 4 ki te \frac{5253}{2}, ka 10506.

$$3x+10506x\left(x+25\right)^{-1}=2\times 300+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te 300, ka 600.

$$3x+10506x\left(x+25\right)^{-1}=600+2x\times \frac{1}{2}$$

Whakareatia te 2 ki te \frac{1}{2}, ka 1.

$$3x+10506x\left(x+25\right)^{-1}=600+x$$

Tangohia te x mai i ngā taha e rua.

$$3x+10506x\left(x+25\right)^{-1}-x=600$$

Pahekotia te 3x me -x, ka 2x.

$$2x+10506x\left(x+25\right)^{-1}=600$$

Whakaraupapatia anō ngā kīanga tau.

$$2x+10506\times \frac{1}{x+25}x=600$$

Tē taea kia ōrite te tāupe x ki -25 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+25.

$$2x\left(x+25\right)+10506\times 1x=600\left(x+25\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x ki te x+25.

$$2x^{2}+50x+10506\times 1x=600\left(x+25\right)$$

Whakareatia te 10506 ki te 1, ka 10506.

$$2x^{2}+50x+10506x=600\left(x+25\right)$$

Pahekotia te 50x me 10506x, ka 10556x.

$$2x^{2}+10556x=600\left(x+25\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 600 ki te x+25.

$$2x^{2}+10556x=600x+15000$$

Tangohia te 600x mai i ngā taha e rua.

$$2x^{2}+10556x-600x=15000$$

Pahekotia te 10556x me -600x, ka 9956x.

$$2x^{2}+9956x=15000$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+9956x}{2}=\frac{15000}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{9956}{2}x=\frac{15000}{2}$$

Whakawehe 9956 ki te 2.

$$x^{2}+4978x=\frac{15000}{2}$$

Whakawehe 15000 ki te 2.

$$x^{2}+4978x=7500$$

Whakawehea te 4978, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2489. Nā, tāpiria te pūrua o te 2489 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4978x+2489^{2}=7500+2489^{2}$$

Pūrua 2489.

$$x^{2}+4978x+6195121=7500+6195121$$

Tāpiri 7500 ki te 6195121.

$$x^{2}+4978x+6195121=6202621$$

Tauwehea x^{2}+4978x+6195121. 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}.

$$\left(x+2489\right)^{2}=6202621$$

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

$$\sqrt{\left(x+2489\right)^{2}}=\sqrt{6202621}$$

Whakarūnātia.

$$x+2489=\sqrt{6202621} x+2489=-\sqrt{6202621}$$

Me tango 2489 mai i ngā taha e rua o te whārite.

$$x=\sqrt{6202621}-2489 x=-\sqrt{6202621}-2489$$