Whakaoti i te 1000x^2-590x-561=0

$$1000 { x }^{ 2 } -590x-561=0$$

Whakaoti mō x

$$x=-\frac{51}{100}=-0.51$$

$$x = \frac{11}{10} = 1\frac{1}{10} = 1.1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 1000x^{2}+ax+bx-561. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-590 ab=1000\left(-561\right)=-561000$$

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 -561000.

$$1,-561000 2,-280500 3,-187000 4,-140250 5,-112200 6,-93500 8,-70125 10,-56100 11,-51000 12,-46750 15,-37400 17,-33000 20,-28050 22,-25500 24,-23375 25,-22440 30,-18700 33,-17000 34,-16500 40,-14025 44,-12750 50,-11220 51,-11000 55,-10200 60,-9350 66,-8500 68,-8250 75,-7480 85,-6600 88,-6375 100,-5610 102,-5500 110,-5100 120,-4675 125,-4488 132,-4250 136,-4125 150,-3740 165,-3400 170,-3300 187,-3000 200,-2805 204,-2750 220,-2550 250,-2244 255,-2200 264,-2125 275,-2040 300,-1870 330,-1700 340,-1650 374,-1500 375,-1496 408,-1375 425,-1320 440,-1275 500,-1122 510,-1100 550,-1020 561,-1000 600,-935 660,-850 680,-825 748,-750$$

Tātaihia te tapeke mō ia takirua.

$$1-561000=-560999 2-280500=-280498 3-187000=-186997 4-140250=-140246 5-112200=-112195 6-93500=-93494 8-70125=-70117 10-56100=-56090 11-51000=-50989 12-46750=-46738 15-37400=-37385 17-33000=-32983 20-28050=-28030 22-25500=-25478 24-23375=-23351 25-22440=-22415 30-18700=-18670 33-17000=-16967 34-16500=-16466 40-14025=-13985 44-12750=-12706 50-11220=-11170 51-11000=-10949 55-10200=-10145 60-9350=-9290 66-8500=-8434 68-8250=-8182 75-7480=-7405 85-6600=-6515 88-6375=-6287 100-5610=-5510 102-5500=-5398 110-5100=-4990 120-4675=-4555 125-4488=-4363 132-4250=-4118 136-4125=-3989 150-3740=-3590 165-3400=-3235 170-3300=-3130 187-3000=-2813 200-2805=-2605 204-2750=-2546 220-2550=-2330 250-2244=-1994 255-2200=-1945 264-2125=-1861 275-2040=-1765 300-1870=-1570 330-1700=-1370 340-1650=-1310 374-1500=-1126 375-1496=-1121 408-1375=-967 425-1320=-895 440-1275=-835 500-1122=-622 510-1100=-590 550-1020=-470 561-1000=-439 600-935=-335 660-850=-190 680-825=-145 748-750=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -590.

$$a=-1100 b=510$$

Tuhia anō te 1000x^{2}-590x-561 hei \left(1000x^{2}-1100x\right)+\left(510x-561\right).

$$\left(1000x^{2}-1100x\right)+\left(510x-561\right)$$

Tauwehea te 100x i te tuatahi me te 51 i te rōpū tuarua.

$$100x\left(10x-11\right)+51\left(10x-11\right)$$

Whakatauwehea atu te kīanga pātahi 10x-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(10x-11\right)\left(100x+51\right)$$

Hei kimi otinga whārite, me whakaoti te 10x-11=0 me te 100x+51=0.

$$x=\frac{11}{10} x=-\frac{51}{100}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$1000x^{2}-590x-561=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1000 mō a, -590 mō b, me -561 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-590\right)±\sqrt{\left(-590\right)^{2}-4\times 1000\left(-561\right)}}{2\times 1000}$$

Pūrua -590.

$$x=\frac{-\left(-590\right)±\sqrt{348100-4\times 1000\left(-561\right)}}{2\times 1000}$$

Whakareatia -4 ki te 1000.

$$x=\frac{-\left(-590\right)±\sqrt{348100-4000\left(-561\right)}}{2\times 1000}$$

Whakareatia -4000 ki te -561.

$$x=\frac{-\left(-590\right)±\sqrt{348100+2244000}}{2\times 1000}$$

Tāpiri 348100 ki te 2244000.

$$x=\frac{-\left(-590\right)±\sqrt{2592100}}{2\times 1000}$$

Tuhia te pūtakerua o te 2592100.

$$x=\frac{-\left(-590\right)±1610}{2\times 1000}$$

Ko te tauaro o -590 ko 590.

$$x=\frac{590±1610}{2\times 1000}$$

Whakareatia 2 ki te 1000.

$$x=\frac{590±1610}{2000}$$

Nā, me whakaoti te whārite x=\frac{590±1610}{2000} ina he tāpiri te ±. Tāpiri 590 ki te 1610.

$$x=\frac{2200}{2000}$$

Whakahekea te hautanga \frac{2200}{2000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 200.

$$x=\frac{11}{10}$$

Nā, me whakaoti te whārite x=\frac{590±1610}{2000} ina he tango te ±. Tango 1610 mai i 590.

$$x=-\frac{1020}{2000}$$

Whakahekea te hautanga \frac{-1020}{2000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$x=-\frac{51}{100}$$

Kua oti te whārite te whakatau.

$$x=\frac{11}{10} x=-\frac{51}{100}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$1000x^{2}-590x-561=0$$

Me tāpiri 561 ki ngā taha e rua o te whārite.

$$1000x^{2}-590x-561-\left(-561\right)=-\left(-561\right)$$

Mā te tango i te -561 i a ia ake anō ka toe ko te 0.

$$1000x^{2}-590x=-\left(-561\right)$$

Tango -561 mai i 0.

$$1000x^{2}-590x=561$$

Whakawehea ngā taha e rua ki te 1000.

$$\frac{1000x^{2}-590x}{1000}=\frac{561}{1000}$$

Mā te whakawehe ki te 1000 ka wetekia te whakareanga ki te 1000.

$$x^{2}+\left(-\frac{590}{1000}\right)x=\frac{561}{1000}$$

Whakahekea te hautanga \frac{-590}{1000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x^{2}-\frac{59}{100}x=\frac{561}{1000}$$

Whakawehea te -\frac{59}{100}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{59}{200}. Nā, tāpiria te pūrua o te -\frac{59}{200} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{59}{100}x+\left(-\frac{59}{200}\right)^{2}=\frac{561}{1000}+\left(-\frac{59}{200}\right)^{2}$$

Pūruatia -\frac{59}{200} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{59}{100}x+\frac{3481}{40000}=\frac{561}{1000}+\frac{3481}{40000}$$

Tāpiri \frac{561}{1000} ki te \frac{3481}{40000} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{59}{100}x+\frac{3481}{40000}=\frac{25921}{40000}$$

Tauwehea te x^{2}-\frac{59}{100}x+\frac{3481}{40000}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{59}{200}\right)^{2}=\frac{25921}{40000}$$

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

$$\sqrt{\left(x-\frac{59}{200}\right)^{2}}=\sqrt{\frac{25921}{40000}}$$

Whakarūnātia.

$$x-\frac{59}{200}=\frac{161}{200} x-\frac{59}{200}=-\frac{161}{200}$$

Me tāpiri \frac{59}{200} ki ngā taha e rua o te whārite.

$$x=\frac{11}{10} x=-\frac{51}{100}$$