1000x^2-590x-561=0 eching

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

x uchun yechish

$$x=-\frac{51}{100}=-0,51$$

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

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 1000x^{2}+ax+bx-561 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

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

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -561000-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

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

Har bir juftlik yigʻindisini hisoblang.

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

Yechim – -590 yigʻindisini beruvchi juftlik.

$$a=-1100 b=510$$

1000x^{2}-590x-561 ni \left(1000x^{2}-1100x\right)+\left(510x-561\right) sifatida qaytadan yozish.

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

Birinchi guruhda 100x ni va ikkinchi guruhda 51 ni faktordan chiqaring.

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

Distributiv funktsiyasidan foydalangan holda 10x-11 umumiy terminini chiqaring.

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

Tenglamani yechish uchun 10x-11=0 va 100x+51=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1000 ni a, -590 ni b va -561 ni c bilan almashtiring.

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

-590 kvadratini chiqarish.

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

-4 ni 1000 marotabaga ko'paytirish.

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

-4000 ni -561 marotabaga ko'paytirish.

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

348100 ni 2244000 ga qo'shish.

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

2592100 ning kvadrat ildizini chiqarish.

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

-590 ning teskarisi 590 ga teng.

$$x=\frac{590±1610}{2\times 1000}$$

2 ni 1000 marotabaga ko'paytirish.

$$x=\frac{590±1610}{2000}$$

x=\frac{590±1610}{2000} tenglamasini yeching, bunda ± musbat. 590 ni 1610 ga qo'shish.

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

\frac{2200}{2000} ulushini 200 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{590±1610}{2000} tenglamasini yeching, bunda ± manfiy. 590 dan 1610 ni ayirish.

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

\frac{-1020}{2000} ulushini 20 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

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

561 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -561 ayirilsa 0 qoladi.

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

0 dan -561 ni ayirish.

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

Ikki tarafini 1000 ga bo‘ling.

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

1000 ga bo'lish 1000 ga ko'paytirishni bekor qiladi.

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

\frac{-590}{1000} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

-\frac{59}{100} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{59}{200} olish uchun. Keyin, -\frac{59}{200} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{59}{200} kvadratini chiqarish.

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{561}{1000} ni \frac{3481}{40000} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

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

x^{2}-\frac{59}{100}x+\frac{3481}{40000} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

\frac{59}{200} ni tenglamaning ikkala tarafiga qo'shish.

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